 Alright so today I'm going to start with one of the most interesting chapters in maths, permutation and combinations. This chapter is going to be very useful not only in the chapter itself but also in probability which you're going to do this year, next year, early in first year of undergrad and this topic will be important also for your KBPY exams. Okay so this chapter is just like you know coding right you have to be not skillful in that but you have to have that art of coding okay because it's based on your principle of counting. Many of us are naturally best with that art of counting. Many of us acquire it through a lot of practice okay so if you are the one who has that art of counting will find this chapter very easy but some of you who are not yeah and some of you who are not that good you may find a little bit of repulsion from this chapter but let me request you this that if you spend a little bit more time on this chapter and practice different kind of problems you soon start feeling comfortable with these concepts okay now before I begin with this chapter permutation and combination itself I would like to talk about certain tools that you must be aware of before we start the process of counting or before we start counting how many arrangements and how many combinations are possible okay so first concept that we are going to talk about is the concept of factorial I'm sure you are aware of this concept correct but just to be formally introducing this concept to you factorial is first of all represented by n followed by an exclamation mark where n has to be a whole number where n has to be a whole number okay how it is defined it is defined as the product from n n minus one n minus two all the way till you reach one okay to give you an example four factorial if I say four factorial is going to be four into three into two into one okay we know we should know the value of factorial still seven at least okay three factorial again it's three into two into one that's going to be six two factorial two into one that's two one factorial one itself what about zero factorial one how do we know that and for the reason of it he said just like that okay now I saw some video it has something to do with like all in progression like the answer to this comes from the pattern that you observe over here if you divide by that number yeah if you see the pattern to get three factorial you divided four factorial by three isn't it sorry four factorial by four isn't it yes I know to get two factorial you divided three factorial by three yes I know if you continue this trend to get one factorial it's nothing but two factorial by two zero factorial is one factorial by one okay that is nothing but one by one that is one itself okay more importantly you need to observe that a higher value of the factorial could be expressed in terms of lower value by using of this relation n factorial is n into n minus one factorial right that means four factorial could be obtained as four into three factorial okay it can further be written as four into three into two factorial further it can be written as four into three into two into one factorial which is nothing but four into three into two into one okay this property of expressing a higher value factorial in terms of a lower value factorial is very very important right are you getting this and this chapter as he rightly says comes directly from looks very simple but when it comes to solving questions on factorials I have seen people making lot of mistakes let me start by asking you this question number one find the remainder find the remainder when summation of our factorial from r equal to 1 to 100 is divided by 15 is divided by 15 please type in your response in the chat box once you're done or you can also speak out unmute your mic and you can speak the answer out sir is there some number never mind any idea one second they're getting something sir after 15 factorial or yeah from 15 factorial onwards 15 will make there's a five factor it onwards right not 15 faculty 15 faculty is too far you since five factorial onwards you'll start observing that okay I'll not say right around let's let us try so who will I've joined in a Kiran is Dirich yes sir okay why so fancy name it's my mom's name my name is my mom's laptop that's what's okay so it's not that difficult once you start writing the summation the first term is one factorial then you have two factorial then you have three factorial four five six and so on till you reach hundred factorial right okay now when you're dividing by a hundred basically you're trying to see what is as we discussed right now divided by 15 not yeah sorry we're dividing by 15 we want to see and as I discussed right now five factorial onwards all the terms will be divisible by 15 that means they will not leave any remainder isn't it because five factorial itself has a three and a five factor in it correct before five factorial that means from one to four factorial these terms may not be divisible by 15 so what I'm going to do is after five factorial including five factorial the remainder is going to be zero so I'm not going to even bother looking at these terms I'm just going to inquire within one factorial to four factorial some correct so one factorial as I already told you is one by the way should know the value of the factorial still seven at least one factorial is one two factorial is two three factorial is six four factorial is 24 five factorial is one twenty six is seven twenty seven is five zero four zero okay so this term which is nothing but 33 okay the remainder that will come will be only because of this term 33 and when you divide 33 by 15 I'm sure you have seen this symbol in your coding okay 33 ampersand 15 that means what is the remainder when 33 is divided by 15 the remainder is three so he answer students so we don't know what any of that means no problem you at least know what is the remainder when 33 is divided by 15 any questions is it clear please type CLR on your screen if it is clear to you okay next is another summation question what is summation of R into R factorial from let's say 1 to 100 find the summation of R into R factorial from 1 to 100 wait thinking no sir I'm just kidding like the long way let's look into this I will be using the property which I just now discussed with you that a higher value higher value factorial can be expressed in terms of a lower value factor okay so what I'm going to do is I'm going to write let's say R it's term over here is R into R factorial correct I'm going to write R as R plus 1 minus 1 R plus 1 factorial we all know is R plus 1 factorial you are it's term is expressed as a difference of R plus 1 factorial and okay now let's factor R value as 1, 2 etc till we reach 100 so when you put 1 you get 2 factorial minus 1 factorial 2 will be 3 factorial minus 2 factorial t3 will be 4 factorial minus 3 factorial and so on if I reach till t100 correct it becomes 101 factorial 101 factorial minus 100 factorial let us add them let us add if you add them then your sum will be okay all these will get cancelled 101 factorial minus 1 factorial that is nothing but 101 factorial minus 1 okay that's going to be your answer in general I can say in general I can say summation of R into R factorial from 1 to n is actually nothing but n plus 1 factorial minus 1 okay just remember this result sometimes it comes directly as a question in comparative exams okay if possible I'm not saying that you have to remember this in case you want to save your time just remember this now something which is slightly tricky but I want to see how much you can actually work out this work on this particular problem find the exponent of 300 factorial find the exponent of the word exponent 3 power something find the power of what power 3 is raised by a hundred-factor brilliant absolutely correct be this so if let's say 100 factorial where to be prime factorized okay I am I will not write all the prime factors whatever what is going to be the power on 3 that is what it is asking that's the meaning of find the exponent of 100 factorial so how long do we get for this I mean right now I'm going to give you around three minutes you should be able to solve it within 30 seconds okay one second then okay yes sir one minute yes sir I'm getting something okay just just write out just write out the name is to help you to think is not to solve problems for you okay if I'm able to successfully make you you know think in different variety of problems then I would think my job is done sure 48 48 you know what you are correct very good I was not expecting this to be very fine but well done you have solved it okay come on Dheeraj 54 okay so first let us try to see how many trees can we extract in the first pass say all the terms which are multiple of three like three six nine twelve fifteen dot dot dot it will actually contribute one three each correct correct now from three six nine twelve fifteen eighteen twenty one twenty four etc till hundred of in fact till 99 can I say that I can extract 33 trees directly can I say I can extract hundred by three gif trees so so many trees can be extracted from these numbers which I have encircled with yellow yes or no yes sir yes or no okay I'm not writing all the terms because of course it would be too long to write all the terms okay now can I say there would be certain terms which even after losing a three will have one more three in them for example nine is one of them do you agree with me or not yes it twenty seven is another one correct can I say some terms would contain more trees even after taking out one three correct every factor of nine yes so can I say if even if I take out every factor of nine that number of trees will be collected so as you rightly said Dheeraj those numbers would be actually hundred by nine gif these square brackets are not ordinary brackets they are gif okay now despite taking out two trees there will be still some numbers which will say three terms especially like 27 will be contributing 81 will be contributing so can I say the quantity of such trees that I can extract would be this yes or no okay and it will continue the same trend till I start realizing that all the terms which were supposed to contribute a three has been fully extracted out of them three has been fully extracted out of them so after I think 81 which gives you one three trees there will be no other terms which will contribute in fact you can see the term if you write 100 by 3 to the power 5 they will all start vanishing they'll all start becoming 0 0 etc okay that means the number of trees that I will get will be 33 plus 11 plus 3 plus 1 we counted it is 44 plus 4 which is nothing but 48 trees can extract from 100 factorial that means this question mark here is 48 okay now by looking at this expression by the way this expression we normally name it as e 300 factorial this is a convention that we follow this is a convention that we follow if you want to find out what is the exponent of any number n or any number p in 100 in let's say n factorial we would use the notation e that number and the factorial of that now I'm going to give you a important formula over here the formula is finding the exponent of finding the exponent of a prime number in n factorial okay the formula is written as gif of 100 by p n by p 100 was still in my mind let's say 100 by p gif of 100 by p square gif of 100 by p cube keep on doing keep on following this pattern till you start realizing that your answer is giving you 0000 okay one important thing to be kept in mind over here this formula works only when you are finding the exponent of a prime number p doesn't work if the number is not prime are you getting this point why it doesn't work when the number is not prime because let's say the number was 6 6 could come not only from various powers of 6 but also from numbers like 12 right which doesn't appear as any exponent of 6 isn't it that that process of finding the exponent of a non-prime number would be much more complicated however do we break it into prime right if you if you want to find the exponent of a non-prime number just find out the exponent of those prime numbers which are contributing towards making that number and find the least of those exponents i'll take an example based on that one second sir please uh please note this down this is a very important formula this is called the lujan's formula lujan don't pronounce it as legendry okay okay waiting the is prime it's called lujan's formula yeah lujan's uh question yes sir find the exponent of find the exponent of 80 in 180 factorial what what that's find the exponent of 80 in 180 factorial oh is there time one no one minute sir no sir one and a half minutes so it's three right three all right well let me think it should be three right lujan's formula it doesn't work for a non-prime number huh so prime number yeah sir that's what i'm doing no sir i know it's not a prime number but 80 can only come wherever there's a factor of no whenever let's say from eight factor no yeah like that it still will come up no from eight factorial no it'll come up sir sir you have a doubt you need four twos and you need one five right yeah yeah i think in terms of getting the two and five from 180 factorial yeah sir what's the doubt uh just since the time is less i'll just tell you what i'm thinking okay uh it's like we'll do the operation for both the numbers for both the prime factors and whichever is the lesser thing that'll be the power because whatever i was saying whichever is the limiting reagent sir chemistry sir knowing the chemistry not already 180 by five less oh and in this case a limiting reagent is five anyways but remember the exponent of five will be lesser than exponent of two for sure yeah number four moles of two reacts with one mole of five to produce one mole of 80 so you have to take that ratio also into account are you yeah sir i came here to take a break from chem please yeah i think you have to form a compound which is called 80 so one mole of 80 takes four mole of two and one mole of five one second i wait madi sure plus seven plus one two 44 sir 44 is absolutely correct see adhira jadweta see how many twos you can extract and how many pies you can extract that means first find out what is the exponent of 180 factorial okay let's use the lejean's formula 180 by two gif 180 by two square 180 by two cube 180 by two to the power four 180 by two to the power five 180 by two to the power six 180 by two to the power seven i think beyond this we need not go because 180 by two to the power eight will become 256 and that will start giving you zero zero zero okay so this is 90 this is 45 this is 22 this is 11 five to one if you add it you'll get 176 okay similarly let us find out the exponent of five in 180 factorial so that will be 180 to the five get 180 by five square 180 by five cube i need not go any further because five to the power four then start giving me zero correct so this is going to be 36 plus seven plus one okay now this is 44 correct now thankfully this is say exactly in the same ratio as what we wanted right because this is four it to 44 right correct so to make one mole of 80 we need one more more more and they're exactly the same proportion so we'll have 44 as our answer but let's say hypothetically hypothetically let's say this was only 160 then what would have been the answer can you tell me three right right 40 would have been the 40 i find the ratio 40 yes yes because every four of two will combine with one of five to produce an 80 so since there are 162s you can only form 42 to the power fours that is 40 16s can be formed okay so even if five is more in number it doesn't matter the limiting reagent is basically your reagent number two sir could you just repeat that formula sigma one i equal to one r is sigma that first thing i'm just repeating it's in because i think i might have written it wrong what one sigma from i is equal to one to r r into r factorial 1 to n 1 to n yeah so can you just show it there oh one second which board was it on one i guess yes no it was it was on the third board this one yeah yes sir i is equal to equal to one to one to n minus one yes sir okay yes i got next very interesting question find the number of zeros at the end of hundred factorial okay find the number of zeros so in other words find the power of four at the end of fact what we are saying here is in other words just find how many two into five there are right what is the exponent of ten in hundred factorial that's one minute to do this 24 that's correct so one second i'm getting stuck at addition one second sir almost done Venkat what did you say was the answer 24 yeah he's very simple you want to find out how many zeros are there at the end of hundred how many tens are getting formed right in order to find how many tens are getting formed you need to see how many twos and fives are getting formed but you know that if the power of them are same the limiting reagent will be five isn't it this would be the limiting reagent correct so your number of zeros is nothing but number of zeros is nothing but it's the exponent of five in hundred factorial that's it okay let's use the lujan formula okay i think i don't need to go further yes sir this this person actually proved it sir which person this person only yeah yeah lujan yeah yeah you got the formula see even i did something like very great about it okay now i'm going to start with something which is going to be your core principle behind solving this chapter and do the fundamental principle of costing one two three four fancy numbers fancy names now why they call fundamental because they are so fundamental that you cannot prove them let me ask you this simple question okay let us say bank that has gone to watch a movie mission okay please complete the cinema hall okay there also is some with some books and all okay now let's say this cinema theater has got three rooms sorry three doors on his right and four exit doors on his left okay after finishing the movie how many ways can he exit the cinema theater the three doors on this side and four doors on this side in how many ways can he exit the seven seven ways what did you do with three and four add them you added them what was the reason for adding it why didn't you multiply why didn't you raise three to the power four why didn't you do four to the power three why didn't you do fourth root of three because they're not i mean there are not so many ways you can't say there are four into one point seven ways of exiting the thing yeah so explain me the why why do you do not multiply it why you just added it because you can just see the possibilities and there's no other rule it's just you can just exit from one door and counting it yeah you're fundamentally just counting the way that's use the word fundamentally right so basically you are saying that i could exit the when it could exit the cinema from this door or this door or this door or this door or this door or this door the moment you're saying or or or logically it means you're adding that okay more in general if there is a task if there is a task which could be completed by doing any one of these subtas let's say t1 t2 etc till t r r some subtas any one of these subtas if you do your job will be completed okay for example in this case the task of pink it was to get out of the cinema theater so subtas first taking root number or gate number the first gate here okay or he could have completed the task even after even by getting out of the second gate or he could have completed the task by getting out of the third gate the task is completed by doing any one of these subtas independently there's no dependency on the other subject then the fundamental principle of addition says fundamental principle of addition says number of ways of doing the task would be nothing but the number of ways of doing each one of these subtas so you just add the number of ways of doing these subtas to get the total number of ways of accomplishing that task is that fine yes now a similar situation now let's say this time bank it is outside the cinema theater now his task is to get inside and turn exit yeah by using these entry gates let me call it as en1 en2 en3 you can only enter the cinema hall a cinema theater through these entry gates and there are four exit gates on this side okay ex1 ex2 ex3 ex4 okay tell me in how many ways can he get in and of course you'll see this time the answer is three into four okay why you multiplied here because for every entry he has got four exit options so for three entry options he will have three into four which is nothing but 12 exit of 12 parts correct so here you can say he can enter through this and exit through this he can enter through this and exit through this he can enter through this and exit through this like that you can keep making different different parts now here your job is your task is made up of two subtas entry and exit the number of ways of doing entry is three number of ways of doing exit is four so fundamental principle of multiplications if there is a task which is made up of these subtas let's say t1 t2 till tr and all the tasks all the subtas must be completed in order to complete that task and to say that the number of ways of doing the task would be the product of the number of ways of doing these subtas okay so here i am going to be three into four that is 12 ways to do it so could you just go up a bit yes sir uh no a bit more up here yeah sir 11 once you can sit down t r uh yes sir these principles are so fundamental that uh no we don't care much about it but i tell you within the heart of problem solving of pnc why is this fundamental concept now there is one more principle of founding is for the principle of inclusion and exclusion that i will discuss later on with you as of now i would like to answer a few questions i would like to answer a few questions okay let's let's see first question is first question is there's a station master who has been provided five colored flags so five colored flags have been provided to the station master okay to make one signal to make one signal the station master needs at least two flags the station master needs at least two flags okay find the number of signals that he can generate how many different signals can he generate remember even if he's even if he's using two colors uh let's say he is using let's say the five colors are with good why violet indigo blue green yellow okay so what's five let's say these are five colors so if violet is first and let's say green is second this is this will be considered to be a different signal as compared to green first and violent second okay please yeah diraj you were saying something do you have an answer no no no i was asking what five factorial is five factorial no no i'm asking what five factorial is 120 thank you sir is it 10 one second sir then i think i made some mistake oh sorry 60 60 okay others i want everybody to respond here thanks sir one second sir it's a lot it's a lot okay now see if you want to create signal the signal can contain two flags okay or your signal can contain three flags or your signal can contain four flags or your signal could contain five flags am i correct okay now there's a reason why i wrote or or or because you should know fundamentally what operation you're going to perform okay let's focus on first making a two flag signal now in order to make a two flag signal what is the number of ways you can choose the first flag here i don't know you guys mute yourself yeah so in order to choose the first flag how many ways can he make that choice he can make that choice in five ways because he has been provided with five color flags correct and he has to choose the second flag also but given that he has already used one color for the first flag he has only got four options left correct so for the second color he can choose the second color in four ways correct now remember he has to choose both the first flag and the second flag that means fundamentally what operation do i have to perform between four and five i have to perform multiplication operation getting my point or means plus right remember whenever you are using the word or whenever you are using the word cases you need to add whenever you're using the word and you need to multiply okay can we now address the number of ways of making a three flag signal so in the three flag signal the first flag you can choose in five ways second four ways third three ways again plus this you can make five four three two again a plus then five four three two one basically you have to add all these values which let's see what does it come out to be 20 and this is 60 and this is 120 and again this is 120 so your answer is going to be 320 ways he can generate a signal is that fine everyone understood your mistake everyone yes sir yes sir okay yes sir here okay let me ask you another question let's say diraj wants to post a letter okay he has got seven letters okay i don't know whom he has addressed the seven letters but he wants to post them and he sees four letter boxes he sees four letter boxes okay how many ways can he post these letters given that there's no restriction on how many letters he can put in a given letter box it's a very simple question i hope you have understood the question he can post any number of letters in any letter box he wants he can put all the seven in the first letter box also is it a very huge number sir you don't have to calculate it just tell me the expression is it four into seven factorial no wrong sir it's something like what we did last time just instead of the amount of flags it's the amount of post boxes kind of sir is it four parts seven four to the past seven okay please don't hesitate in giving out your answer it may be wrong doesn't matter yeah i'm just trying one second i think as if every letter has got options okay so let's say letter number one how many options it has four or correct it can fall in any letter box letter two also has four options letter three also has four options letter two letter seven that will also have four options every letter here has got four options correct now my fundamental question to you is what am i going to do with these seven fours do i add them or do i multiply them you multiply them because it's or no you can go into any box you multiply them that means you should be and right and means i said i said or or it doesn't mean multiplication add means or means that's all i say multiplication no let me ask you this question if you post your letter number one is your job over no i have to post all seven so you have to post letter one and letter two and three and letter four tell you your task was to post all the letters you can't complete your task just by posting one letter correct yeah sir yes so what's your answer yeah four into four into four etc till four that is nothing but four to the power seven did you say that Venkat second time sir second time are you getting this point sir can this be thought in a way like you toss in two coins and both the times you'd get one like that number of outcomes you're tossing a coin coin finding the number of outcomes right tossing not one coin two coins three coins four coins and then and then thinking of the number of time number of ways combinations you get get but the outcomes only two yeah in this case you can think like there's a coin which gives you four outcomes correct yeah yeah same thing process is analogous to that yes sir okay now i'll ask you a similar question let me see how many of you able to answer this there's a lift okay this lift stops at eight floors it's a flow number one two three four five six seven eight okay that's the number of people in the lift this four contains five people okay this four this lift contains five people inside the lift okay how many ways can these five people get down on these eight floors eight power five power five why not five to the power eight i don't know sir just came into my head like see sir it can't be it i mean it is eight to the power five not five to the power eight because yeah because one guy has so many options and that the floor can't get into the lift yeah correct okay every person here has got eight options eight options eight options eight options eight options it's not necessary that the one first should have only one person getting down on it okay but if i make a restriction that let's say one person can get down only on one floor that means one floor can only accept one person is on it then how many ways by the way so this is eight to the power five now i'm saying that on a floor only one person can get down that is the floor can't have two people right sorry the floor cannot have two people no okay so then one second let me think sir yes not sure but i don't know just feel yeah eight into seven into six into five into four that's correct why are you hesitant in answering so first guy will have eight options the second guy can only have seven options because he cannot choose the floor where the first guy has gotten correct similarly the third will have six options fourth will have five options and finally the fifth guy will have just four options so that is nothing but if you write this properly it is actually eight factorial by three factorial am i correct am i correct yes sir yes sir on we learn that this term is actually called eight p five later on we learn right now you don't need to worry right now focus on what was the logic involved are you getting my point okay one more question i would like to ask you let's see how many of you are able to answer this let's say there are um four rings okay and we have five fingers okay cannot draw as good as the resist but i've tried my best in how many ways can you put these four rings in these five fingers does that you can put any number of rings in any finger you want and these are all different rings remember the way you are putting a ring that sounds very different for example if you put ring one first and ring two then it is different than putting ring first and ring one then next so to the power four five to the power four okay and what one answer i mean it is five to the power four minus or no it's five to the power four plus one second let's think um it's five to the power four plus some or it's more than five to the power four more than five to the power four it's a 120 120 okay now who are choosing their fingers or the rings the fingers i mean it's it's like the the people in the elevator are like the rings people in the elevator are like the links so people are the choosers right so rings are the choosers yes right ring can choose that's a ring number one ring number two ring number three number four right ring number one has got five options this one has four options is correct no this also has five options why it can't go into the same finger very okay again answer is five to the power four okay somebody said four to the power five i guess no sir five to the power five to the power okay see what what is helping you to solve this problem there's no formula per se to solve it is just right connecting to the basics of the fundamental principle of counting let me introduce two permutations now basically we are done with the basic tools that is factorial tool and we are also done with the concept of fundamental principle of addition and fundamental principle of multiple there's one more principle which is all principle of inclusion and exclusion which i i'll deal with you later on sir how much of the space required for that inclusion thing like how many lines approx uh leave two pages excluding questions only formula like the like fun only formula live around two page two page oh okay okay so first of all permutations what are the meaning of the word permutation itself permutation means arrangements okay when you say arrangements it means selecting things and providing them some order for example if i say how many ways can you arrange two objects taken from three objects so let's say your three objects are a b c okay how many ways can you arrange two objects taken from three objects so you can say this is one way this is another way this is another way this is another way this is another way is there any other possibility no right no sir so basically this is the number of ways in which you can arrange two objects chosen from three objects and where these three objects must be distinct getting my point this is when you say three people a person who reads this he he makes it in his mind that there are three objects given to this person and he's selecting two out of those three objects remember all that three objects must be distinct this formula doesn't work if your objects are identical and this formula doesn't work if uh some of them are also identical okay now from those three objects you are picking up any two and you're assigning order to them are you getting my point in general we use the expression npr if you want to say um please mute yourself everyone okay npr means arrangement of our objects taken from and distinct objects this word here distinct is important taken from and distinct objects okay now obviously a question will arise in your mind what is the formula for npr okay so now let us derive the formula for npr so let's do the derivation for the formula of npr okay remember here r has to be less than equal to n okay and of course n and r must be whole numbers okay now when you say npr the image that i would like you to create in your mind is let's say there is a class which has got n people okay and students are sitting in this class okay now the teacher here puts r seats so what she has done she has put r chairs or r seats in front of the class okay now she wants to arrange our students from these n students on these r chairs okay how many ways can she do that let us first find out how many possibilities are there to fill the first chair uh you can now unmute yourself and speak out how many possibilities are there to fill the first chair and and how many possibilities are there to fill the second chair and and minus one okay the person who has always sat he can't sit there also yeah i cannot sit on the second chair also okay this is not modi's argument okay by the way there is a joke which i want to crack at this moment of time yesterday was mr modi's happy birthday right so rahul gandhi says happy birthday to mr modi so mr modi says thank you then rahul gandhi says uh party kaha hai modi says humari to har jeega hai tumari kaha hai to translate it he says that he has got his i understand that much in the sir i know that okay in how many ways can he fill that third chair n minus two correct fourth chair n minus three now look at the pattern and tell me very very carefully very very carefully how many ways can the teacher fill the arath chair n minus n plus one n minus r yeah n minus n plus one n minus r one r minus one yeah correct now can i say according to fundamental principle of counting i have to multiply all of them why not add all of them so when you're saying add because it's not or yeah if you're adding means we're trying to say that even if the first guy sits the job is done no right you have to make the first guy sit and you have to make the second guys second guy say and third guy and fourth guy till you reach n minus r plus one correct this is your answer but generally this expression is too long and ugly to remember so what do we do we do some bit of no simplification of this formula what i'm doing is i'm going to multiply i'm going to multiply the next term that would have come in this series by the way what would have been the next term after this had this series continued n minus r n minus so what i'll do i'll multiply and divide with n minus r in short i'm not doing anything to the expression okay whatever i'm multiplying with i'm dividing it also with the same expression correct yes or no yes sir now what would have been the next term n minus r minus one n minus r minus one so i'm multiplying down also with n minus r minus one next term n minus r minus two i'm multiplying down also with n minus r minus two if i continue doing till i reach one that means i can continue doing this till i reach one correct oh now we all know that from n if you go all the way till one what is it called in fact that's why these tools were introduced because it it actually makes our representation of expression very very simple down also you'll have n minus r factorial correct and here comes the formula for npr please remember this you are going to see this formula for the next one and a half years yes sir npr is n factorial by n minus r factorial so a quick thing for you what is np n and what is np zero calculate fast fast fast it should not take you more than five seconds n factorial this is the formula is n factorial by n minus r factorial right and one right one this is one absolutely simple you have to just use n factorial by n minus n factorial over here which is n factorial by zero factorial which is nothing but n factorial by one which is a whole factorial okay this is nothing but n factorial by n minus zero factorial which is nothing but one okay now this is a important thing which says that if you're finding factorial of a number it means the number of ways of arranging n objects taken all at a time are you getting my point for example if i say how many four digit how many four digit word can you make from one two three four zero how many four digit number oh number can you make from one two three four what you'll have to be all the objects and arranging it among themselves right the different way of arranging will create a different different number isn't it yes or no one two three four that's one number two one three four another number so what are you doing you're just shuffling the position of these numbers four one three two another number like that so like this can create 24 numbers are you getting my point is that clear yes sir this is another way of saying four p four also correct yeah if you want to go more fundamental you will say hey i want to make a four digit number no problem let me choose this digit in four ways but once a digit is chosen please note that the repetitions are not allowed allowed repetitions are not allowed let me write it down the next i can choose in three ways this i can choose in two ways this i can choose in one according to the fundamental principle of counting since i have to choose the numbers on the hundred space sorry thousand space hundred space ten space one space multiply all of them and i guess my answer is 24 not choose the way you want to you know find out your answer normally i always choose the most basic manner or most basic way of sorting the problem because these formulas will start failing once the problem becomes complicated to never forget your basics all these formulas they are coming from your fundamental principle of counting correct they have not come from the sky getting my point so if you remember your fundamental properly all these formulas can be generated j and all these comparative exams which are of serious nature they focus on whether you understand the evolution of those formulas are then knowing those formulas getting my point chalo i'll make your life a bit complicated if all the four digit numbers that you have formed by using one two three four repetitions not allowed if they're added up what will be the sum so what all the four digit numbers that you have made by using one two three four there 24 such numbers right yeah if you add them all what could be the sum one plus two plus three plus four are you not one one two three four right other number let's says four one two three oh okay like that if you keep adding if you add them all what is the rate sir sir how much time and two minutes how much two minutes 120 seconds when good okay fine wait sir some thing 24 numbers no i'm not adding four four sir if i'm not wrong the last digit there are six numbers where it'll be four six numbers where it'll be two six numbers where it'll be something something like that right okay thank god the unit number will be six into four fact no sorry it'll be um six into four plus three plus two plus one yeah that's what six into one plus two plus three plus four which is whatever six into ten yes yeah i know it and zero six into ten and then the next one also be six into ten but you have to just um carry the number yeah six into ten plus six so 66 six into ten plus six six six six thank god is it six six six zero six six six zero yeah yeah tag team wow yeah nice stop shouting that's getting record where did sir go i don't know they'll probably i i was muting and talking uh sorry uh do both of you concur on that answer yes sir six six six zero i think others where are you kindly participate okay please mute yourself so that i can explain this now both of you have figured out very correctly there would be six such numbers which will end with a one correct so such numbers would be six in number correct because you can choose the other in three into two into one way so six is the number of ways similarly six numbers will end with a two correct similarly six numbers will end with a three and finally six numbers will also end with a four correct yes or no okay so this accounts for total 24 in count now if you just add your unit's place if you just add your unit's place the sum of all the numbers in the unit's place will be one plus two plus three plus four and each are occurring six number of times isn't it because there are six ones six twos six threes six fours so that is nothing but six times ten which is nothing but 60 similarly the sum of the number in the tens place will also be 60 okay in hundreds place will also be 60 okay and in thousands place that will also be 60 okay the sum of the numbers will be 60 but if you want to find out the actual sum units are assigned a place value of one tens are assigned a place value of 10 hundreds are assigned a place value of 100 and thousands are assigned a place value of 1000 okay so this will be your sum of all the numbers right let's add them and see how much does it come out to uh nostalgia so six zero zero zero and then you have six zero zero zero then you have six zero zero then you have six zero so your answer is six six six six zero that's your answer is this what you got yes sir yes sir awesome very good okay now few questions first i will take on your understanding of this formula itself npr is n factorial by n minus r factorial so let's say we want to take some questions based on this questions based on this formula easy ones only i'll give you first question is first question is if if uh 56 pr plus six is two 54 pr plus three is three zero eight zero zero is two one find r sorry find r rp two let me make slightly different questions find rp two the time your time three minutes let's start three minutes is also on the higher end but try to complete it any idea but sir wait sir one second sir doing something sure one second sir so what's 56 into 55 what do you want so that was a calculation 56 into 55 3080 80 oh you did that mentally sir yeah at two 51 minus r is equal to so one six four zero one six four zero is absolutely correct okay awesome thank you sir okay let's let's try to solve this so 56 pr plus six and i did have 56 factorial divided by the difference of these which is 50 minus r factorial is two means divided by 54 factorial and the difference of this is 51 minus r factorial correct this is zero eight zero zero okay now 56 factorial can be expressed in terms of 54 factorial like this 56 into 55 into 54 factorial and 51 minus r factorial could be written as 51 minus r into 50 minus r factorial we'll see how many terms would end up cancelling up here so 55 56 as we have discussed is 3080 54 factorial gone 50 minus r factorial gone 51 minus r is equal to 10 that means r is equal to 41 if r is 41 your answer is going to be 41 p2 41 p2 is nothing but 41 factorial by 41 minus two whole factorial 41 factorial by 39 factorial which is 40 into 41 which is six one six four zero guys let me tell you one important thing here don't don't start making your own formulas with factorial i've seen people doing this m plus n factorial m factorial m factorial m factorial into n factorial please there is no such formula like this okay so don't start making your own formula now before i move on to the other problem there is some property which i want you all to hold on properties of mpr everybody please mute yourself npr as i already told you the first property that was npn is n factorial and np0 was 1 okay next property is you could write npr as n times n minus 1 pr minus 1 can you prove this by logic i know with the user formula you can prove it but i want you to prove this by logic no formula i don't want to use npr formula i don't want to use n minus 1 pr minus 1 formula i just want you to give me a simple storyline and prove that npr is n times n minus 1 pr minus 1 sir it's like that four things only like let's say we have four options yeah you're just reducing an option and yeah you're reducing an option and one denominator also like one customer one shopkeeper both of them are reduced and then you're and you're keeping that outside right like four into three into two into one see again you are using some kind of formula right four into three into or two into one i don't want you to use any formula sir if there's three people and then this there's three people and i'm sorry yeah there's like three people and three doors consider my story well you'll get a hint of what i'm expecting from you let's say there are n people okay and we have r cheers in front of us correct so r cheers n people and you want to arrange r people chosen from those n people on those r cheers correct that is what we call as npr correct now what i'm saying i can break this task as first choosing a guy who will sit let's say this n how does this n come one guy one person okay let's say one person you choose from this correct yes hello let's say one person you choose from this how many ways can you do that ten ways and yeah correct now this fellow he has to be seated on these r cheers correct correct let's say you make him sit somewhere on these r cheers correct now how many people more are supposed to be selected from how many remaining n minus one n minus one people are remaining yes sir correct and you're selecting how many more from them uh r minus one r minus one so the number of ways to arrange that will be n minus one pr minus one yeah yes or no correct so basically what you're trying to say is that there is one special chair let me call it as the hot seat okay and first you're selecting a candidate for this hot seat and that you can do in n ways because you have been provided with n people and you have to choose one person from those n people to make him seated on this hot seat that is n ways remaining r minus one cheers you have to fill from n minus one people right so the number of ways you can arrange r minus one people from n minus one people is n minus one pr minus one why this multiplication because unless until both the activities are done unless until both these activities are done my job of making or my job of arranging our people on uh my job of arranging our people on sorry n people on our chairs will not be accomplished correct so these are used to sub tasks am i clear okay yes so no formula here use just logically i'm trying to prove it why i'm so keen on using logic is because later on we need to develop such thinking abilities to solve complicated problems if you don't start right now it'll be too late getting my point yes sir just take the third property okay by the way this can this property can be scaled up say you can say n n minus one into n minus pr minus two then you have n n minus one n minus two into n minus three pr minus three and it can keep on going okay till wherever you want to third property n minus one pr is n minus r times n minus one pr minus one fourth property n pr is n minus r plus one into n pr minus one okay now you don't have to remember these properties can anybody prove the third one just for logic this is the same as this so just move the fourth one one second sir yeah sir from r minus one won't um the last move the fourth one by using logic you do yourself when you have to speak it's like you remove one chair that is one possibility one option for all the students to sit see left hand side is arranging our people chosen for yeah one second just think about it i mean there's no hurry we can solve questions meanwhile also okay just get back to me once you've figured out the reasoning we'll discuss it let me take a problem first 22 pr plus one is to 20 pr plus two this is equal to 11 is to 52 find the value of r just three minutes to solve this once you're done please feel free to speak out the answer or type in your response in the chat box please wait two minutes sir just a just a small help for this uh no i'm not asking just asking sir like seven into six into five we can write as seven factorial by uh five factorial right seven into six into five we can write it as seven factorial by four factorial okay so the outermost number manage the number of numbers by once i can say 21 minus r minus three 21 minus r p 15 minus r wait manage it minus r factorial by minus factorial uh one second sir this is some new thing sure we got some new thing see the ideal is you give the second thing you should do but not figuring it out you guys any idea almost sir i think i know okay one just a second sir 52 sir like let's say we have this we need to subtract such a number from 21 nothing sir why are you doing that you can use formula here right yeah that only i simplified it okay not that much this is 11 by 52 correct okay by the way 20 factorial you can cancel from 22 factorial yeah it'll become just 22 into 21 and even this will get cancelled too this will be one correct so two into one is 42 into uh 18 minus r factorial by 21 minus r factorial you could write it as 21 minus r into 20 minus r 19 minus r and 18 minus r factorial that would get cancelled with this correct hmm okay and i'm sure all of you must be getting this uh expression 20 21 minus r 20 minus r 19 minus r is equal to 42 into 52 okay there is product of three consecutive numbers is that absolutely you don't have to solve a cubic equation to get r you just have to observe here that it's a product of three consecutive numbers so this also you have to break it up as a product of three consecutive numbers how will i do that let's see 42 you can write it as 16 to 7 uh 52 you can write it as 4 into 13 correct now break this 4 as 2 and 2 correct attach 1 2 to this attach 1 2 to this so it'll become 12 14 into 13 oh make it as 12 13 14 if you want okay yes sir just compare this with these three numbers all you just comparison and you can do any one of the comparison that means you could compare either the lowest with the lowest or second lowest with the second lowest or the highest with the highest any one of them will give you the same answer so let's say you do the comparison of 19 minus r by 12 r will become 7 this is where it's getting stuck sir yeah so in case if this was like really hard to do would you recommend just doing cubic roots no no it'll always be some kind of product of consecutive numbers you don't have to go to cubic to solve it okay getting the point okay next i'll take some word problems determine the number of permutations you can make from the letters of the word please mute yourself everyone determine the number of permutations of the letters of the word simple done taken all at a time taken all at a time what do you mean by that like find the different number of permutation of the letters of this word taken all at a time that means how many how many words are there how many little alphabets are there one two three four five six eight nine correct so how many nine alphabet words you can create form okay okay is it nine factorial yes sir nine factorial that's it how much time it takes answer is nine factorial or you can say nine p nine same yeah yeah nine p nine okay so sir should we uh can you just write nine factorial or do we have to solve the whole thing no no you can write nine factorial but you have to give an explanation in your school exams the number of ways of arranging nine objects chosen from nine distinct objects is nine p nine which is nine factorial is that fine yes next question is find the number of five digit numbers made from one two three four five okay find the number of five digit numbers made from one two three four five which are divisible by four divisible by four we can't end with one three or five yeah exactly just find all of them and then it's um two by or three by five of that no later the edge one second trying one two three two five four is it two into four factorial one second sir one second sir one second sure is it four factorial four factorial that's correct how did you do that uh my logic was that um if you find all the possibilities out of that only two by five of them will end with an even number and out of that half will uh be factors of four because every odd um not every alternate factor of two is a factor of four one second sir i'm i did it in two cases one second just like two minutes ish sure sir sir and i got 24 but in a different way i guess branches 24 you may get it in a different way i think you would have done the cases wise right yeah when two comes only one three or five can come in the 10th place and four only two can come in the 10th place okay and then uh and remaining back you did it just do some juggard and then get the number of think here also do some juggard get the number of digits and then uh do that okay so i have uh a suggestion for all of you can you take those cases where see but when does the number is when when is the number divisible by four when the last two digits are correct yes or no so my last digits two digits could be 12 24 32 or 52 yes sir now if the last two digits are fixed the remaining three digits could be filled up in three into two into one way for each one of them yeah these are your four cases case number one case number two case number three case number four and each one of them is going to give you six plus six plus six answer is 24 done enough so what i did i just fixed the last two digits of these five related numbers so last two digits could be 12 could be 24 could be 32 could be 52 now if you have 12 the other three can be shuffled among themselves in three factorial days yes or no three factorial plus three since cases are there you have to add them so three factorial plus three factorial yes sir we talk about those cases where you are permuting from things where certain things are identical so let's take that cases where we are trying to find the permutations of objects with identical objects of some type so let us say we have n objects okay so why i'm writing factorial m objects okay out of these n objects there are p objects which are identical of type one identical of let's say type one there are q objects which are let's say identical of type two and let's say there are r objects which are identical of identical of type three okay then how many ways can you permute these objects among one another or how many ways can you arrange these objects by taking all the objects at a time let me give you a simple example to relate to it let's say the word Mississippi correct if you see this word Mississippi there are four s's in it correct one two three four okay there are two p's in it correct and there are four i's in it one two three four okay so altogether there are how many words one how many alphabets one two three four five six seven eight nine ten eleven so your n is eleven here okay your p you can say is four q you can say s2 and r you can say s4 now my question is how many 11 alphabet words can you make from the letters or from the alphabets of Mississippi how will i find this please note that my answer will not be 11 factorial or my answer will not be 11 because all my 11 alphabets are not distinct yeah there's so many same here okay to answer this i will start with a simple question so that you are able to scale it up to a complicated concept like this let's say i talk about the word moon okay and i say how many different four letter or four alphabet words can you make from moon okay now remember two o's are identical over here they are identical so you cannot say your answer is four factorial that would be wrong right but let us say for the time being i assume them to be different let's say o one and o two just like your isotopes of oxygen correct so how many different words can you make from it then you'll say four factorial correct but remember for every word that you make let's say you made a word o one m and o two okay it's the same as o two m yeah what you considered was these are two different words according to your answer of four factorial but in reality they are the same word because o one and o two are same correct that means for every such case your answer has become doubled correct yes or no yeah that means you need to divide your answer or discount your answer by two or you can say two factorial to get to the original answer so this will become your answer so the number of words you can make number of four alphabet words you can make from moon is only four factorial by two factorial are you getting this point now i'm going to generalize this if you want to have a combination uh sorry if you want to have a permutation of n objects out of which p are identical of type one here type one is you can say sss are type one ppp are type two iii are type three like that then your number of ways of arranging these words would be nothing but n factorial divided by the factorial of the repeated words are you getting my point and you can scale this up depending upon how many reputations are there in your question is that clear second sir please could you just wait sure sir if it wasn't for something that you identify with itself like you you have an r also it would just be a n p f r by this thing right no n p r cannot be used at all if n if it's not that's why these formulas become useless when your problem starts becoming complicated sir sir asked like how many six letter words can you make with this or something complicated question those questions i'll take up a little later on as upon the basics here remember as diraj rightly pointed out here we are using all the letters taken together all taken at a time if the number is the if you make it more complicated let's by saying you are only taking four letters out of it or six letters out of it that would become more complicated and those type of questions i'll be taking up a little later on yes sir so this chapter can be complicated like anything right yeah yeah yeah okay let's take a question how many words can be formed by using the letters of the word using all the letters of the word intermediate there is a 12 factorial over um three factorial into two into two factorial there are three is no yeah this is a two 12 factorial by three factorial into two into two factorial because there are uh two t's and two i's there is two t's also no yeah okay got it yes sir one yes sir one two three four five six seven eight nine ten eleven twelve eleven factorial by two only are repeated can i see i can see two i is repeated so two factorial how many e's are repeated how many e's are repeated just becomes your answer yes sir how many words can be formed with the letters of the word partly putra without changing the relative position of the vowels and consonants partly putra sir i got how to do it just one second there's no meaning of the relative position of the vowels and consonants don't change there is there only a yeah wherever a vowel is there only a vowel can come in the new word right right so for example the first alphabet p here could only be replaced with the either a t or a l or a r like that okay sir i spelled the word wrong one second sir basically you take each vowel as one type and you take each consonant as one type yeah spelled the word wrong sir i got how many vowels are there wait accounting wait wait wait we 11 factorial by five factorial into five factorial hey the bottom has to be five and six what bottom has to be five and six if on the top it's 11 it can't be if you just take the number in front of the exclamation mark it has to be equal no because there's no letter that can't be a vowel or consonant what oh i think i counted wrong one second one yeah i counted oh yeah yeah yeah 11 factorial by six factorial into five factorial what i did i separated the consonants and vowels okay now remember there can be arrangements of vowels among themselves and consonants among themselves right correct so in how many ways can you arrange the consonants among themselves now remember there is a repetition of two p and two t's so can you tell me how many ways can you arrange these six alphabets where two p's and two t's are repeated can i say six factorial by two factorial two factorial how many ways can these consonants sorry how the how many ways can these vowels be arranged 20 we'll say five factorial by three factorial correct what do i do with these two numbers do i add them or do i multiply them multiply absolutely because the consonants and the vowels have to be arranged the moment you use the word and that means multiplication so the answer will be six factorial by four into 120 by six how much is it can you just calculate 20 and i think this is 120 30 one seven uh oh okay sorry sorry yeah can i say it is 3600 yes sir one second just check if mine also will come to that Venkat will what we found also come to this sir i don't think so because you got 11 right and 3600 sorry 11 won't get cancelled in the denominator and 3600 is not a multiple of 11 so our answer is wrong next question insurance how many ways can the letters of the word insurance be arranged so that the vowels are never separate huh so what does that mean when the vowels are never separate means all the vowels must be together oh okay so nice one 20 60 into 2440 sir yes is it 1440 1440 is not correct hold one second one second 84 84 yes sir 84 is too less too less it's good only sir i guess yeah sir is it um i don't know how but i think it's six factorial into 12 six factorial into 12 that's correct okay okay so how do we do this see if you take all the vowels i hello yes hello Venkat okay okay i thought sir was explaining yes so i u a e these are your uh vowels correct okay and what are the other alphabets n s r n s r n c right correct so this is a method that we normally use here which is which i call as the string method what i do for string methods i actually tie up the letters which i want to be together with a string right correct and treat this as a special alphabet we can see that we have one two three four five six alphabets now let me put one okay so these six alphabets can be arranged in six factorial ways oh and the vowels also can be arranged among themselves yeah i'll come to that uh Venkat oh yes sir there are two repetitions so you have to divide by two factorial and you rightly said these vowels can be intra arranged can be intra arranged getting my point okay so ultimately the answer will be six factorial into as we rightly said into 12 which is actually seven to zero into 12 which is six thousand eight thousand six hundred and forty is that clear yes sir so why is it written on all four sides of the screen but how do you interact with that screen you have an annotation oh request remote control oh no i'm not i'm not okay got it oh this is a question this is the first one find the number of combinations that can work omega one second sir two and two three factorial so is it two and three factorial you mean to say 20 sorry 12 yeah they cannot they can only change their own positions correct so only they can be shuffled in two ways or two factorial ways the remaining three here that is MEG they can be shuffled among themselves in three factorial the two factorial three factorial answer is 12 is is the next one yeah four factorial i got it please feel is the next one two factorial oh no one second i need to wait sorry sorry it's not two factorial anyway i tell you third one is the third one is always in the middle so you cannot shuffle can i say rest four can be shuffled in four factorial ways so answer is 24 yes sir 12 12 yeah three fours occupy odd places so wobbles this is only the case where wobbles can occupy odd places so you have oh yeah okay and they can shuffle among themselves in three factorial and m and g can shuffle among themselves in two factorial so answer will again be 12 correct wobbles never being never together one second sir isn't that just the same as wobbles occupying odd places that's absolutely correct answer for the fourth one will also be 12 these two are same questions actually okay i will take up some questions which are on arrangement of words in a dictionary which we call as a dictionary questions so i'll start with a question let's say we have a word here Krishna it's actually Krishna okay now if this word where to be arranged in all possible manners that means if you form all possible six alphabet words from Krishna remember no repetitions are allowed the words can only be repeated as many number of times it is actually sorry the alphabet can only be repeated as many number of times as they are present in this particular word okay that means all of them cannot be repeated actually okay because every word is present every alphabet is present only once okay now my question to you is if you write the alphabets if you write these alphabets in all possible manner and arrange them as in a dictionary arrange them as in a dictionary all of you know how dictionary the words are written alphabetical order correct what would be the rank of what is the rank of the word Krishna itself in that dictionary sir what's the value of the six factorial 720 sir is it 240 plus 3 into 4 factorial do you understand the meaning of the question first of all yeah so take a simple word cat okay if i write all possible three letter words the first letter would be act then atc then you have c at then you have cta then you have tac then you have tca correct in that in this the rank of the word cat is the third position this is at the third position same thing i want you to do with the letters of the word Krishna in that you have to find the rank of the word Krishna so tell me the rank it will be a position let's say you say 377th word is Krishna like that i i go i need the number and scout is it the 312th position okay 312th what about others that's not right then i'm not sure if it's right but is it 317th position 317th okay i'm waiting for others to answer girls you're all quite i don't know why write out let it be wrong sir yes could you tell if this is right or not i'm making some mistake here in there like i don't want to proceed because the calculation becomes after this too much after this like you start with a and then you find out the number of combinations of permutations for that then you start with i absolutely you are on the right direction then k a k i then k r correct does that give that does that does this give you a hint girls how to begin with so basically what banker is saying he's saying that all the alphabets will start from all the words will start from a okay let me first write down the very first letter in the dictionary so what i try to do is i try to find how many of the words that begin with a or i which is okay and then i added that to how many ever i took k out and within five how many started with any letter that is before r and then third before yeah yeah same methodology so basically this is your first word in the dictionary right they cannot be any word before this correct correct now what diraj and menkut both are saying that all the letters all the words which begin with a okay so let's say i fix the first alphabet to be a here okay so all the words which begin with a how many words will begin with a remember the five of them are free to shuffle around so five of them will shuffle around in five into four into three into two into one which is actually five factorial so start using factorial now since you have already learned it can i say these 120 words will definitely be above krishna right so if you want if you want to find the rank of krishna you are actually counting how many words are before krishna isn't it similarly all the words which begin with i they will also be of krishna so can i say those numbers will also be five factorial correct but when you come to k you have to be careful because krishna also begins with k so you cannot blindly say all words which begin with k will be above krishna right so we have to choose the next in line so the second in command is k a correct can i say all words which begin with k a will be above krishna and how many such words are there which begin with k you'll say four factorial because the last four can be shuffled among themselves in four factorial ways am i correct is that correct okay next would be k i this will also be four factorial okay next would be k in this will also be four factorial next would be kr but when i come to kr i have to be careful because krishna so i have to choose the next in line kr a so can i say three factorial such words would be there correct then kr i now i have to be again be careful because krishna also starts with kr i so i have to choose the next in line kr i a so two factorial such words r i no not r r is taken r is before us no no r is taken r is already taken so i can choose r again correct k no k is already gone sorry kr i n okay then kr i s what happened how does suspense somebody asked how does in between and stop i think my brother i'm sorry got it okay now with k r i s the first word will be krishna that is number one and finally krishna that is one more okay let's add it okay so five factorial plus five factorial how much is it 120 okay 120 plus 72 sorry 240 plus 72 3 12 3 12 3 12 plus 6 3 18 3 18 plus 4 3 22 3 23 3 24th word is krishna this is your answer i was close i just messed up the last part you know your closeness would be like this your options will be 3 22 3 23 3 24 3 25 they're looking as they're looking at the mistake where we'll count the rank wrongly right yes yeah these type of questions are very favorite you can come in your school as well as competent exams oh yes then okay now let me take a question where again there's a dictionary question but some alphabets are repeated for example let's say i take a word suriti okay if the alphabets of the word suriti are arranged in all possible manner and return in a dictionary what would be the rank what is the rank of the word suriti in that dictionary sir yes we have i two times so we take it once right right yes okay but not always you have to see where you're taking it actually like sir what she said will make a difference when it comes to suriti and suriti that stays right since when you are trying to shuffle the two eyes but not make a difference if you are starting with that later okay you first give it a try then we'll discuss it so is it 827 there's less than 720 words how can it be 827 oh what is uh oh okay okay one second six factorial is 120 right six factorial is 720 oh yeah so yeah just a second sir i think i made a small mistake just a second there already adding left two two okay so 143 143 okay sure done anybody who's done done anybody could do this okay let's see okay let me solve this first see first of all if i write the very first alphabet it will be i i r s t u correct this is the very first word correct now if you want to find out the rank of the word suriti can i say all the words will start with i okay can i say all the words will start with i will definitely lie above the word suriti correct remember since one eye is locked there's only one eye which is free to move okay so let's say if i lock this word sorry this alphabet the alphabets which i see they're all different right they're all distinct i r s t u okay so they can arrange themselves in five factorial ways yes or no now you don't have to lock another eye because this eye will take care of all the eyes okay so you can directly start with r now because the next inline is r okay many people do a mistake they put the second eye also just like the previous question correct now if you if you fix up this r now the remaining five alphabets will have two eyes repeated correct so if you just hide lock this r you have i i s t u correct how many five letter words can you make from i i s t u you would say five factorial divided by two factorial why divided by two factorial because there's two repetitions because your two eyes are repeated is that clear okay next comes the word s okay now when s comes i have to be careful because suriti also starts with s okay the next is s i how many words will start with s i remember now one eye is locked so only one eye is available so that can be shuffled among themselves in four factorial ways is that fine everyone okay now i don't have to put another eye because this eye takes care of all the eyes so next is directly s r correct s r the number of words that i can make will be four factorial by two factorial why this two factorial because now those four words will have two repeated eyes so you must be seeing that somewhere i'm dividing by two factorial somewhere i'm not dividing by two factorial where am i dividing by two factorial where i have two eyes to shuffle with where there's only one eye to shuffle with i will not divide by anything are you getting my point okay now the next in line would be s t s t now for s t remember again i'll have to do four factorial by two factorial because two eyes are free to move okay next we come to s u but i have to be careful because suriti also starts with s u so i have to do s u i okay for s u i this will be three factorial only because one eye is locked and only one eye is free to be arranged getting my point next will be s u r now remember suriti also starts with s u r so i have to write the next in line which is i okay again suriti also starts with i so i again have to write next in line which is i then only only only one word is in form which is sur i i t okay and finally s u r i just shuffle the position of the suriti okay let's count so let's count 120 this is 120 this is 60 this is 24 this is 12 this is again 12 this is 6 plus 2 more okay how much is it 180 plus 24 is 48 48 plus 8 which is going to be 180 plus this is going to be 56 so 236 the word is suriti is it clear yes sir okay one more in the dictionary question alligator oh again okay tell me what is the 49th word when you arrange the letter the word again in a dictionary so if you make a dictionary by if you make a dictionary by arranging all possible permutations of these five letters okay and put them in a dictionary what would be your 49th word in the dictionary 432 sir naik naik okay so give me two minutes yeah yes sir naik naik okay let's discuss this so can i say the very first word would be agin okay sister of magin okay so if you if you take all the let all the words starting from a can i say that will be four factorial let's keep a count 24 already gone next will be g g will be factorial by two factorial because two a's are free to be shuffled so that is going to be 12 so 36 already gone okay so it will be nagi not naag any g double i nagi where did i get the double i from no n double a g i she means oh yeah correct correct right g comes before i i forgot letters also nagi sister of magin okay next is with i i will again be how much is it now 48 is already 48 just cite the next word in line the next word will be the very first word that begins with n so a a a g i this will be your 49th word this out of curiosity what will be the 50th word half century word na naik only these two will shuffle this will be your 50th word okay but now i'm going to begin with a new concept that's the concept of circular permutation now what's the circular permutation and how it is different and why do we need a special attention to this let me illustrate this by asking you a simple question let's say there's a round table i'm dieting here and there are people who want to sit around this round table to have a discussion correct in how many ways can they be seated around this round table in how many ways can abc be seated around this round table the six wrong remember the order of this there's no starting and end of a round table okay that's the important point that we need to consider if you start shuffling them all let's say you put c in the place of a a in the place of b b in the place of c okay i'm sorry b in the place of c these two arrangements would be counted as same correct even if you shuffle it once more let's say let's say now c comes in the place of a a comes in the place of b b comes in the place of c they were counted as the same they are not different sitting arrangements they are the same sitting arrangements guys remember one statement of mine in case of circular permutation okay if you want to have a different sitting arrangement then the relative partner of at least one person should change so could you repeat okay i'll write this down in circular permutation in circular permutation a different arrangement is obtained is obtained if the seating position i can say the relative seating position of at least one person changes but here the seating positions of none of them are changing for example let's say c was to the right of a and b was to the left of b sorry c was to the right of a and b was to the left of a correct similarly a was to the right of b c was to the left of b so a is on the left of c and b is on the right of c in all these positions if you see they have the same relative positions with respect to each other so they're just changing their chairs just like we play that game of you know what is that game we should play musical chairs musical chairs yes right not exactly but everybody is shuffling one one position they would be all counted as the same sitting arrangement are you getting my point so what will possibly give a different sitting arrangement in this case let's say if a was sitting here and b was sitting here and c was sitting here then this would be different are you getting the point these all are same same same same okay can there be any other position remember if you stop the position of a and b then it is as good as a b c it'll actually be this position getting the point so my dear students here the answer is there can be only two sitting arrangements n factorial by n i guess exactly and that's what i was going to do next i'm going to generalize this so basically if you had n persons or n objects of course distinct objects to be arranged in a circular permutation in a circle in a circle okay then the number of ways to do that will be n factorial discounted by n why discounted by n because all n shufflings that they will have will be counted as the same correct see what i'm trying to say is that let's say p1 is sitting here p2 is sitting here okay p3 sitting here like that i break i go up till pn okay okay let's say i take a scissor and i cut the circle over here okay and i open it up as a linear permutation so if i write it as a linear permutation it'll be p1 p2 p3 up till pn okay okay now remember even if i make if i make the cut at this position that means p it starts with pn pn p1 p2 etc till pn minus one okay even if i make the cut over here that is pn minus one pn p1 up till let's say p i think what will come here pn minus two right okay note that all these will be counted as the same arrangement but in a linear permutation they will be counted as different right so if you consider not to be a circle then your answer will be n factorial but remember n of these arrangements would be the same so you have to discount your answer by a factor of n that's why your answer will actually become n minus one factorial please make a note of this getting the point now here one thing you should not forget see the formula is not going to help you everywhere one thing you should not forget is it's basically the shuffling of them in n seats which gives you the same arrangement you have to divide by that n that is important that means unless until the relative partner of at least one of them change it will not be counted as a different sitting position is that clear yes sir if that is clear let's take up a question sir my apartment's lift sir oh my god every centium class i got that yeah it's open or what lift is always open do this sir who's that speaking in the background here at sir hey i'm asking sir who's speaking in the background that's what dad here at sir is speaking in the background oh i thought he meant diraj comma sir no sir i asked who's speaking in the background and he said diraj sir i thought he thought that you were asking him who was speaking in the background please wait sir thinking think sir yeah it's a kind of a weird answer 12 factorial plus 13 plus 14 plus 14 24 can you use factorial notation to make it look simplified yes at 12 factorial plus after that it's plus adding i don't know if this is right sir but then i thought if there were for 13 people and 13 chair it'll be 12 factorial see but then if there have to make it that complicated you just assume the fact that you have to choose 13 people out of 12 people 24 people how many ways can you do that oh you have to arrange let's say you have to arrange 13 people from 24 people how many ways can you do that 24 p 13 correct now the total number of seats are 13 okay so all the 13 ways of shuffling would be counted as the same so this is your answer oh okay done sir is this also correct like technically what did you do first uh for if there were 13 chairs and 13 people it'll be 12 factorial yeah but if i add one more guy there'll be 13 more combinations that take place that's a i think complicated way of solving it yes sir and if i add two more people first 13 plus another 14 will be there not 13 so is this correct just to check my understanding that was right or not for understanding factorial plus sigma 14 13 to 24 no yeah see they can be they can be multiple ways to do it but is it feasible to solve it by that method just ask in our decision add 11 no sir correct that will not be feasible sir are you going to kbpy now or hey wait i'm going to continue with this chapter because this is also a part of kbpy okay because hey fine hey just 45 minutes more yeah 45 marks in chem will go but i'll sit in no i'll sit when is your chem exam monday right day after oh day after anyways this will be recorded so no problem even if you back or doesn't matter i'll go after a bit leave okay is this clear right yes sir yes i am alone right this one that's all one work in our school sir boys and girls sit different places in how many different ways can five boys and five girls form a circle so that boys and girls alternate sir we should tie them up right no need to tie them up yeah oh okay we shouldn't tie them up i thought he meant literally tie them up first like first see first what will you do you will first arrange the boys and girls let's say i arrange the boys okay ram sit down arm and let's say parshuram i'll say abhiram one two three four five okay can these guys shuffle them can these guys be arranged around the circle yeah can i say four factorial ways this because they're free to move right one one place they can move in fact any kind of rotation will be counted as the same okay now the moment the boys have made themselves seated okay how many ways can i make the girls sit now the girls can only occupy these spaces what i'm showing with x mark again four factorial now bheeraj is saying four factorial but let me tell you bheeraj now let's say i take the name of these girls let's say i call this this girl as sita okay sita rita geeta babita and let's say mita right okay okay now if you are saying four factorial that means dheeraj you are saying that even if sita comes here rita comes here geeta comes here babita comes here and mita comes here they would be counted as the same sitting arrangement i don't think true because earlier sita was sitting between ram and sham now she will be sitting between sham and gansham are you getting five factorial in that case your answer will be into five factorial because the moment you have made the boys sit the circular permutation is lost the circular permutation is lost it is not as good as a linear permutation yeah this is a very good question and this has come a lot of times in the comparative exams which one sir this type of question competitive competitive exams now let's talk about those circular permutation where garland or necklace is involved okay now remember garland and necklace cases are slightly different now let me give you an example and tell you why it is different let's say i want to make a necklace with a diamond a ruby and a talk voice okay now we have only seen that if we shuffle them like this it will be all counted as same right let's say if i make talk voice here diamond here ruby here or let's say ruby here talk voice here diamond here all of them are same correct but let me also tell you that even if you do this diamond here ruby here talk voice here this will also be the same why because you have just taken this necklace and flipped it like this you've just flipped it and kept it flipping a necklace or flipping a garland doesn't give rise to a new garland or a new way of you know placing the you know stones on it isn't it so just try you take your you know whatever i mean let's say you take a necklace which you have you just flip it and keep it does it make a different arrangement of these the gems which are there on that necklace no right it is the same arrangement okay so in this case for every arrangement that you have made flipping is going to give rise to the same arrangement of those stones on that necklace so your answer that we had that is n minus one factorial will further get divided by two so the number of ways in which you can arrange beats around a necklace would be n minus one factorial by two right you understand why this two factor has come because flipping doesn't give rise to a new arrangement right which is different from the case of people so if let's say these were three people then this would have been counted as different isn't it these two sitting I meant would have been different had they had there been people sitting around the round table but now it is not different is my explanation clear to you all yes sir yes sir sir i'll be leaving now if that's okay sure thank you sir give your 45 marks yeah not 45 all right i'll try sir okay thank you bye thank you why sir are you you're continuing right yes sir then even I leave okay okay thank you sir thank you one and a half minute like excluding the question the question time starts after one and a half minutes why I just go to washroom and come back down music because the hero has returned yes sir just now i'm starting sir sir 18p 12 by 24 sir same principle as the previous one yeah divide by 12 and half that also absolutely correct so 18p 12 by 24 next question sir three into 19 factorial three into 19 factorial three into 19 factorial i think you are correct so out of 21 18 is there and three different pearls are there let's say diamond ruby dock wise cut them up and make as one stone correct all together you have 19 correct 19 uh tough uh possible correct now 19 stones you can arrange in 19 minus one factorial correct 21 um uh yes sir correct divided by two because a necklace out of it two three factorial because these three stones can be inter arranged in three factorial ways yes sir getting my point yes sir so your answer should be slightly different three into 18 factorial 18 factorial yeah counting i mean okay yes sir so now i'll begin with combinations combination also means selection remember you are just choosing you are not assigning any order to the chosen ones you're just choosing and you're stopping the process there that is combination correct so if i ask you there are three objects all distinct in how many ways can you choose two objects from there so of course you'll say let's say abcr your three objects then ab ac bc that's it okay now don't start assigning order to a and b don't say ab and ba because the choice will ultimately be the same that is of a and b okay yes sir team okay now if you assign the batting order it will become arrangement getting the point yes sir in order to express the number of ways of choosing our objects from and distinct objects we use the notation ncr many books will write it as n r without a c term like this okay if such a term is seen in any bush please take that as ncr oh i've seen this somewhere sir i wasn't able to solve this okay remember ncr will only work when all the objects are distinct it doesn't work if some of the objects or all the objects are identical okay yes sir i have a question for you prove that ncr formula is n factorial by r factorial n minus r factorial just by logic you derive this formula just by logic given that you know npr formula npr is n factorial by n minus r factorial derive this sir we'll first start with npr itself okay so in ncr itself we have n factorial by n minus r factorial it's npr so what i've done is i've taken all the things i've given them order also okay but i don't want to happen so what i'll be i'll give the order to dissolve them like do whatever you want guys so in that process i'm removing our factorial the our factorial arranging them i'm removing that also see when it is simple way to look at it when you say npr means you are choosing first yeah you're choosing our objects and you are assigning order order to those our objects correct yes sir when you choose our objects that itself is ncr correct yeah and once you have our objects you are assigning order to them so and means multiplication oh r factorial assign order to them in r factorial means correct okay directly gives you the formula of ncr as npr by r factorial which is nothing but n factorial n minus r factorial r factorial oh yes sir okay cool now before i move on uh to give you any other thing i like to discuss the properties of ncr the very first property is nc0 and ncn both are equal to one next property is if ncx is equal to ncy it either implies x is equal to y or x plus y is equal to n okay yeah if you look at this scenario some books also express it as ncr is actually ncn minus r where r is greater than n by 2 logically prove this sir we'll just a second note it down like for logically so like logically speaking right uh n minus r is equal to n minus r and n minus off n minus r is n is r sorry is r sorry uh and and in that place so if you do that in both the cases r and n minus r factorial will come in the bottom like in the denominator is that correct sir just a second just give me a second oh sure sir yeah what you're saying tell me again uh sir um when you do n minus r you get n minus r and n minus off n minus r is that is the formula i was saying logic see this means selecting r people right oh that way you want it sir okay r from n so this means selecting n minus r which you don't want to select okay so if i want to make a team of let's say two from five people right it's as good as choosing the three people whom i don't want to be in the team then automatically yeah people are all selected right so five c two is same as five c three so nc r is same as nc n minus r is that fine but why r greater than n minus two n by two let's think sir because if r is less than n by two then what will happen then this term and this term must be the same sir if i say and if i say five c two five c x then either x can be three or x can be two itself yeah so if it is n by two then this term would be three actually correct okay uh no sir if r is not greater than n by two then what will happen these two terms should be the same then these two terms should be the same okay oh okay yeah yes i got it got it yes sir next property which is the most important one ncr minus one plus ncr is n plus one cr this rule is called the pascal's identity can you prove this logically one second sir ncr minus one plus cr because n plus one cr what does this mean so it means selecting r minus one let's say you want 11 players you take 10 people out of 20 and 11 players out of 20 it'll be like taking 11 out of 21 right that's some that sounds correct okay i'll explain this in a very simple and lucid manner let's say there are n plus one student yes sir and you want to select our students out of it okay how many ways can we n plus one cr now out of this n plus one let's say there is a guy called mr v yeah now let's say i am the selector and uh i have a very good reputation with v so what i do i'll say okay v will always be selected okay don't worry so okay hello yes sir yes sir if we selected then i have to select r more people sorry r minus one more people from n people oh yeah so that's what mr v is selected then i have to select r more people from n people which can be done in ncr way ncr is let's say i'm angry with v i think i have to go to selector oh okay got it sir sorry r minus one no please not selected then i have to select r from n to n minus one oh yeah got it sir yeah that will be into no a plus plus plus plus yes sir got it the white is called the pascal's identity is because parts of 11 day yes sir parts of 11 how are these terms made yes sir you add like uh one and two will form three like only one will come one one and two will come to three if you add these two it will give you three and normally this number here zero sees zero because it is in the super throw and zero zero throw and zero column oh okay this is the row position this is zero throw this is first row this is the second row and the number are in columns zero zero zero column first column second column like that yes sir this number is one c zero zero this is one c one again this is two c zero this is two c one this is two c two again this is yes sir from this formula that if you want to generate let's say this number which is here what is the sum of four c uh yeah the four c two is generated by adding three c one and three c two right yes sir so this is your n r minus one n r this is the n plus one n and r oh okay c r you got it sir just a second sir sure nice so where do we use this triangle actually like other than that binomial expansion the other users also we'll discuss it later on okay sir yes sir problem number four n c r can written as n by r into n minus one c r minus one or you can also see this as r n c r is n n minus one c r minus one can you prove it logically hmm yes sir i can do this just a second sir thinking sure uh sir yes uh let's say we have the same thing uh n students like n students are there are required okay uh and you have one student the n at the guy and in the choosing process you had uh r ways of selecting him like he was the n at the guy okay so but then since you're removing him you need to discount that but then he's still a part of the selection process okay so that's why you're doing or no you're you're doing and you're doing this guy and the rest so i have a better storyline okay yes sir you want to choose our people are from n okay and make a president out of it yes sir one way to do it is i first choose our people and and means multiplication out of this our people how many are eligible to become a president one so our way to make a president our second is i choose a president only first that is anyways to do it oh yes sir and then since president is only chosen i need r minus one more people so both will ultimately lead to the same and result yes sir which is like oh yes sir got it now this fourth property which i'm going to write again this is scalable that means you can you need not stop here you can further write it down as n n minus one by r r minus one n minus two cr two cr minus two you can further scale this up n n minus one n minus two by r r minus one r minus two n minus three cr minus two okay keep some work okay next is n cr by n cr minus one is n minus r plus one by r this formula is very important because it is useful also in binomial theorem oh okay sir just a second sir yeah and so on i'm noting the diary that's why i'm writing it me okay so how does this come uh you tell me a reasoning for this okay next one n c n n plus one c n n plus two c n all the n plus r c n is n plus r plus one c n plus one this is called the hockey stick identity hockey stick yeah why sir i'll tell you first you prove this you can prove it by using any one of the properties we have learned so far okay sir one second n c n just a second sir sir is it just a extended version of formula three n cr minus one plus n cr is equal to n plus one cr yes absolutely it's like for everything you add you keep on adding uh one one one and when you read the rth term you add one after adding r to n correct see basically what are you going to do yeah i'm going to write this term as n plus one c n plus one does it make a difference both are one actually at the end of the day correct yes sir many add these two you get n plus two c n plus one correct do you get n plus three c n plus one correct keep on doing it did you read n plus r plus one if you add these two you get n plus two c n two now if you look at your Pascal's triangle ten ten five one this means you're changing your rows but you're not changing your column okay now see it if you add all of them it actually becomes this one plus one plus one plus one plus one five okay if you add all of them it actually becomes this okay if you add all of them it actually becomes this if you add it actually becomes this like that looks like a hockey stick it looks like a resemblance of hockey stick. Yes sir. 1, 1, 1, 2, 1, 1, 3, 3, 1, 4, 6, 4, 5, 6, 7, 8, 10, 11, 12, 13. Yes sir. Got it? Yes sir. Now let's take few questions on them first. Yes sir. Okay let me begin with this question. Simplify this. Who? Okay sir. So many of the terms converge to be the same. Just a second sir. Especially the last thing. Just a second sir. Last step. Who? Sir is it should be expressed in terms of C and all or actual number value? No, no. Just in terms of C. Okay. 47 C4 plus 51 C4 plus 6 into 56 C53. If that was the case, I would have just written it up. You just have to state it as a C, X, C, Y term. Okay. State the possible values of X and Y. Oh. Okay. One second sir. Yeah, sure. 17, 25, 18, 28, 47, 18 into 20. Right. Just a second sir. I think I know something. Seven. Just almost done sir. Okay. So the answer is 4, 5, 5, 9, 8, 5. You don't have to calculate the value. You see. By the way, do you realize that 56 minus K C53 minus K is always the same. It's just this. Yeah. Why? Because of the use of the property NCR is equal to NCN minus R. Yes sir. So if I keep on changing your K from 0 to 5, okay. Let's say I put the highest value 5. Can I say I'll get 51 C3. 51 C3 yeah. 53 C3, 54 C3, 55 C3 and 50. 56 C3. What is 57 C4 plus 57 C3? What is that formula? 48 C4. Where is that? NCR minus 1 NCR. Oh yeah. Yes sir. Okay. Yes sir. Okay. Now these two will come mine to give you a 48 C4. 48 C4 yeah. 48 C4 and 48 C3 will come mine to give you a 49 C5. Okay. No, no. 4 only. 4, 4, 4. These two will come mine to give you 50 C4. Yes sir. These two will come mine to give you 51 C4. These two will come mine to give you 52 C4. These two will come mine to give you 53 C4. These two will come mine to give you 54 C4. These two will come mine to give you 55 C4. These two will come mine to give you 56 C4. And these two will come mine to give you 57 C4. Okay. What I was looking for is this answer 57 C4 or say 57 C4. 53 yes sir no why I told the actual number of values because I didn't get this so I just calculated how many ways 36 36 how did you do that sir I frankly I didn't use any formula I just did like let's say he's inviting one guy then eight ways for two guys seven ways three guys six ways if he's inviting one guy it will be 8c1 if he is inviting two guys 8c2 or he invites three guys or he invites four guys or invites five or he invites six or invites seven or he invites eight okay now by the way this is related to one of the properties which I wanted to I wanted to learn from this problem okay there's a formula for the formula here is 2 to the power n okay now how do we prove this this is very important this will be useful later on also okay how do we prove this if you have n objects let's say o1 o2 o3 till like that o n objects what does this mean you are selecting none or more objects from it right correct Venkat yes sir that means every object will have two two two options oh either to get selected or not to get selected selected not to get selected correct that means two to the power now how is this helpful in solving this question so each friend will have a choice to attend or not right so if you see this I want the sum from 8c1 till 8c8 if I just include an 8c0 then my answer would have been 2 to the power 8 2 to the power 8 correct but since this term is missing okay let me write down its value it's actually 1 so can I say this will be 8c1 plus 8c2 all the way till 8c8 that would be nothing but 2 to the power 8 minus 1 that's 56 minus 1 which is 255 base okay got it sir got it sir I'll just note it down sir sure oh yes sir don't see yes sir okay so Venkat we'll stop over here sure sir we'll meet next time with more more concepts on permutation in combination yes sir could we solve some kvpy je questions and all yeah yeah sure we'll do that as I'm and sir can we also have some problem solving sessions at school or someplace we'll have it okay sure sir thank you sir thank you for my thank you sir thank you sir thank you