 So as you all know today, we are going to start a new topic and that topic's name is permutations and combinations permutations and combinations Okay, a very important topic not only for J exams But also if you are a KVP why aspirin right because this particular chapter is based on counting skills and it is considered to be a part of your mathematical aptitude Okay, so it basically tests your skills of counting and Therefore, this is a very hot topic if you are a KVP why aspirin or if you are a J advance aspirin or If you are an Olympia aspirin yes, so PRMO RMO people This topic is very much asked Now as I told you this topic is based on your skills. That's why it comes directly from God's book Okay, because some of you may be blessed with that art. Some of you may not be okay in my Teaching span of almost 12 years. I have seen some people can count like this right lightning fast and some people like they're wondering How how how is this like this? How is this like that? Okay? So they would be in senses where you will feel that things are not clicking that fast as Compared to what it is clicking to your friends. Okay, don't get disheartened You can always Win over it through a lot of practice through a lot of practice This topic is useful not only in the topic itself, but also in chapters like binomial theorem Right, so we'll be doing binomial theorem also in some days in some weeks and it will also be useful in probability right probability is what probability is Probability of an event is what number of favorable events divided by the sample space, right? So when you count those favorable events and when you count those sample space You basically need that counting skills and who will provide that skills? This chapter Okay, so three chapters are directly linked to this So I would request you all to be very very attentive and understand the new answers of this Okay So let us start. How big is this chapter? I think three classes minimum minimum three classes if I go to a very Great depth probably take four classes, but I would try to wrap this up in three classes Okay Useful for statistical analysis not really Not really No, I don't think so Okay, yes, if you say if you say probability distribution functions, yes, PNC would be useful there and Probability distribution function is yes remotely linked to stat statistical analysis So you may say yes, there may be indirect connection, but not directly, but not directly Okay, so before I start this chapter a quick introduction about these two words What what do these two words mean? Of course combination is something which you keep on hearing Permutation, permutation means arrangement Permutation means arrangements Okay, when you hear of the word arrangement basically arrangement is a two-step procedure One where you do selection and Whatever you have done as a selection you order them Okay, so arrangement is basically selection multiplied with Ordering, okay, so you do a selection and then you order those selected objects that is called arrangement To give you an example Let's say how many arrangement of two alphabets you can do from these three alphabets So you'll say AB and then you can arrange AB as BA also AC CA correct BC CB BC CD, okay, so six arrangements are possible. You can say six arrangements are possible of Two alphabets from these three alphabets. Okay, so there is a selection Basically, you selected two at a time because my question was in how many ways can you arrange? Two objects or two alphabets chosen from ABC So you selected those two alphabets and whatever you selected you ordered them, right? So that is that would be studied primarily under permutation. What is combination combination is just selection Okay, so as you know the first step of arrangement is selection So if you stop there that means you select and stop do nothing after that that means there is no reference to ordering Then basically that is called combinations, right? So example here I would say if you want to select or if you want to do a combination of two objects or two alphabets from ABC You will do as AB. Now AB BA are considered to be same selections, okay? So there's no point changing their order AB AC and BC. So only three Combinations will come up from it. Okay. Now many people ask me a sir arrangement is Selection and ordering so we should study selection before we study ordering arrangement, right? So the chapter's name should have been combination and permutation. Why we call it as permutation and combination? You'll come to know in some time that permutation is very closely linked to what we call as the fundamental principle of counting Okay, so you will be studying two fundamental principles of counting today and Permutation is directly associated to them. That is why we say permutation is more close to actual You can say instincts of counting as compared to combinations That's why we study permutations first and then we study combinations rather than the other way round Okay, so I hope the difference between the word permutation and combination is clear with the simple example Another example that I would give is let us say I want to choose Let's say you are around 27 of you. So let's say I want to choose a team of five players to represent Centreme Academy for a basketball tournament. Does it really happen? No, I'm just kidding. There's no basketball tournament organized by Centreme Academy So let's say if I want to choose a team of five to represent Centreme Academy for a basketball tournament Then in basketball tournament, you don't need an ordering of the players You just choose those five players and that's it. Your job is done. Job is over But let's say if I want to choose a team of 11 to represent Centreme Academy for a cricket tournament Then not only have to choose those 11 players, but I have to also give them some batting order Right that would be studied under permutations. So basketball team you can say will be studied in the combinations But if I want to select a cricket team, I would have to do no permutations. I have to do arrangements also Sure, sure, sure. How many of you are basketball players? I mean basketball players not like self-proclaimed basketball players. Oh, I am basketball players It's like have you played to a school level or district level or let's say State level or something like that Very good. Very good. Nice. Nice. Nice. Good to see all talented people in my class Nice. Good. You should always keep our physical activity open. You never know Right. Okay. So before we start this chapter We are I'm going to introduce you to a very important tool which is called factorial Now I'm going to show you Factorials, I'm sure you would have heard of the name factorial several times Also, we have discussed several things related to factorial. Okay, just a second guys. Just a second. Yeah I'm sorry. Somebody was at the door. Yeah, so factorial Let us understand factorial first and how it can be used and what are the properties of factorial? So factorial of a number n is written as n followed by an exclamation mark And n must be a whole number. So we are we are calculating factorials or we can calculate factorials only for whole numbers How is it defined factorial is defined as a continued product? from n All the way till you reach one so n n minus one n minus two this continued product all the way till one is called n Factorial now, why does important to learn this concept for several reasons? It is directly associated with counting and it will be a very good tool to have if you want to do larger calculations, okay? The physical significance of factorial is let me write it down n factorial basically denotes the number of ways the number of ways to arrange to arrange n distinct objects Please mark all the words which I'm writing number of ways to arrange and distinct objects among themselves or Taken all at a time Okay, so let's say you have three objects and you want to shuffle Their position or you want to order them then the number of ways to do that will be three factorial The example here that I cited for you ABC Okay, if I ask you in how many ways can you shuffle the alphabets ABC among themselves or in how many ways? Can you order ABC choosing all of them at a time? Then you'll say ABC? ACB BAC BCA CAB CBA if you count one two three four five six, right? So this is actually three factorial, okay? As you know factorial definition is n into n minus 1 till when so three into two into one will give you a six Okay So yes, there are a lot of other uses that will be seen in this entire chapter factorial will be used like anything Okay, we'll be using it almost in every problem so factorial has a very important property What are the property of? factorial any factorial if You see let's say I take an example. Let's say I take five factorial Okay, let's say I take five factorial Okay, what is five factorial five factorial by definition? It is five into four into three into two into one, correct if you see these terms you would realize that Till here, it is actually a four factor isn't it right? So can I say five factorial is just five multiplied to the Factorial of the number just less than this that is actually four. So four factorial, right? I could also observe that I can write it like this That means I could write it as five into four into three factorial, right? And I can continue doing it like five into four into three into two factorial like that So this is a very important property in factorial, which basically helps you to express a Higher number factorial in terms of a lower number factorial So I can always express n factorial in terms of a lower number factorial like this Right, this is a very important property which actually helps us to find zero factorial also because zero factorial May not fit into this definition because if you say somebody start multiplying from zero all the way down till one You will say you have gone crazy. How can you go down from zero to one? Okay, because one is higher than zero But thanks to this property that we will be now able to answer what is zero factorial also, right? As I told you I can find factorials for whole numbers Which also includes zero. So if I write n as one I can write this as one factorial as one into zero factorial So from here zero factorial comes out to be one factorial by one one factorial is just a one Because you're multiplying all the way from one to one. That means one only will be there. Okay, so zero factorial comes from here Okay, so Factorial this property is very very important for us to know we can extend this we can extend this to Any smaller number factorial that we desire so we can stop here if you want We can stop here if you want and so on and so on and so on Okay, so this is something which you should note down because we'll be using it this in many many problems Going forward any question anybody I Think it is related to some kind of Not very sure why it is called so but yes, definitely It's a good question to search for why factorials are called factorials something It's it appears something to be like factors or something, but not very exactly sure why it is called factorials Probably I can Google it out if you want Why factorial is called a factorial? Why factorial is Called So okay, so I got an answer from stack exchange. Okay, it is because they say that The all the numbers involved are actually factors of n factorial So that is actually correct if you see five factorial It will have numbers like five four three two one now all of them are actually factors of 120 Now I'm not saying restricted to it. They can be more factors also, right? They can be 20 as a factor. They can be 10 as a factor, but yes, these numbers are factors of this Okay, good So that question was good question, Anurag Yeah See I also get to learn along with you So there are some questions which make me also think and then I look out for that solution and I also learn So you not only I am your teacher, but you are also my teacher in some way. Thank you Okay, so this is about factorial Now few factorials you must by heart it okay by heart it by lung it by kidney it whatever you call it because they're very Very important and you won't get a time for you to sit and calculate them and especially you don't have a calculator So some things which I want you to remember are the following zero factorial. We just now derived It's one one factorial is one two factorial is two three factorial is six four factorial is 24 five factorial is 120 Six factorial is 720 seven factorial is 5,040 beyond that you would not need it Okay, so knowing these is good enough for your day-to-day dealing with factorials Some properties which I would like you to Know be caution cautious against okay a few important Points I've seen in past people misusing these properties number one There is nothing like X plus Y factorial is X factorial plus Y factorial. Please Do not fall into this trap. Okay? Two plus three factorial is not two factorial plus three factorial. You can check it out 120 Okay, so sorry eight. Okay, so don't fall fall for this property Similarly, if somebody says X minus Y factorial is X factorial minus Y factorial, please Do not fall for this not correct Similarly X into Y factorial is X factorial into Y factorial Definitely not correct. So these all properties I'm writing and I'm catching it off because Many of you would have a tendency Somehow to Apply them in problem solving. They are absolutely wrong. Okay. They're absolutely wrong. Please do not use them Okay, same goes with X factorial by Y factorial if at all they are the X by Y is an integer So X by Y factorial is not X factorial by Y factorial Okay, there's nothing to do it with it. Let's say I take X as 4 and Y is 2 4 by 2 is 2 2 factorial is 2 But if you do 4 factorial by 2 factorial, it will be 24 by 2 which is 12. They're not equal. So all these properties are bogus Please be aware of them. Could you go back to the table? Sure mother value that? Yeah, the small table Okay, is this fine now time to take some questions time to take some questions and Let's understand these properties how they work To n factorial. I've seen people doing this to n factorial. They write it as 2 factor n factor No, no, it doesn't work like that. It doesn't work like that Okay, okay, let's start with a question a very simple question Can I go to the next page People who are copying are you done with it? Have you done this chapter in school so that I can afford to be a little bit fast Have you done this chapter in school? Okay, so let's let's take questions. I'll start with this question Let's do only Only question number three and four Solve for X if it is given that X plus three factorial times 2x minus 1 factorial Upon 2x plus 1 factorial times X plus 2 factorial is 7 by 72 solve for X Okay, in YPR you're doing it. Nice Okay, enough Then HSR guys should rock today. We should be able to answer all the questions What about SSRVM and EPS The done other job. I mean is it going on or done? Okay? Okay, third one. What's the answer? Note that when you're solving factorial questions Please do not entertain any such values Which will make any of the factorial notation or any of the term within factorial a fraction or a negative integer Because factorial operation as I told is only defined for whole numbers So any X value that comes out Through your operations and it happens to make any one of the expression Which is subjected to a factorial as a negative integer or a fraction Those X values will be rejected Very good, Aditya. So let us do it because It's a easy question. Now as you can see X plus 3 factorial and there's an X plus 2 factorial So what I'll do I'll write X plus 3 factorial as X plus 3 times X plus 2 factorial, right? It is basically the use of the property where a higher number of factorial can be expressed as a lower number of factorial, right? Now I can also see in my expression that there's a 2x plus 1 and there's a 2x minus 1 factorial sitting over here So I'll write my 2x plus 1 factorial as 2x plus 1 times 2x Times 2x minus 1 factorial like this, right? So I can go down and down till I realize that I'm getting X minus 1 sorry 2x minus 1 factorial Right, so this is how I have to go down. I can only go down in steps of 1 right So now having done that a lot of terms would get cancelled Okay, so I think this term will get cancelled with this this term will get cancelled with this I think 2 will get cancelled here with 36. Let's cross-multiply. So 36 times X plus 3 is 7 times X 2x plus 1 So this gives me 36x plus 108. This is 14x square Plus 7x. So this gives me 14x square minus 29x minus 108 equal to 0 Oh quite ugly terms are there, but I think this is factorizable as minus 56 plus 27 yeah, yeah, that'll work So 14x take common so X minus 4 plus 27 take common X minus 4 equal to 0 So 14x plus 27 times X minus 4 equal to 0 So this implies X could be 4 or X could be minus 27 by 14 But I'm very sure this is rejected because it will make the terms in the factorial as fractions Okay, so the only value accepted is plus 4. So your answer is X is equal to 4 Well done Aditya. Well done Advik. Very good Without much waste of time can be attempt our fourth one also those who have done this chapter you would Be familiar with this expression. What is this expression by the way and this expression? Those have done this chapter in school Yeah, this is Xc2. This is Xc4 Correct, correct, correct. Okay. So the question is X factorial The question is X factorial is to sorry X factorial by 2 factorial X minus 2 factorial Is to X factorial by 4 factorial X minus 4 factorial is 2 is to 1 fine X Is to means you have to divide something like this Very good pun of nice Okay, I think as you have done this in school. So I'm expecting you to be fast. Yeah, okay Very good So X factorial X factorial gone to factorial and 24 will go by a factor of 12 correct and not only that Let me bring the 12 on the other side So simplifying gives me this 2 by 12 will be 1 by 6 now remember X minus 2 is a larger number as compared to X minus 4 Yes, so let me express X minus 2 in terms of X minus 4 factorial So what I'll do is I'll write X minus 2 factorial as X minus 2 X minus 3 X minus 4 factorial And these two terms will go for a toss which implies X minus 2 X minus 3 is equal to 6 now One way to solve this is of course, you can open it up and get a quadratic out of it Another way would be in fact That is a smarter way and that would be very helpful when we deal with Let's say a cubic or a bi-coder take or a pen take kind of polynomial over here If you see here very I mean closely you would realize that these are product of two consecutive positive numbers Correct. So they are positive numbers for sure. Why positive numbers? Why because they were subjected to factorial right? Right? So X minus 2 has to be a positive I can say a whole number positive whole number. This is also a whole number and two product of consecutive whole numbers is giving you 6 That means it is basically only possible when you are equating this to a 3 and this with a 2 Correct. So if you compare this with a 3 or if you compare this with a 2 Right from both of them your answer would be the same which is 5. That's another way to solve it In this case, of course, it's too easy for you to expand it and do it as well But let's say if it was a Cubic or a bi-coder take where factorization or you can say getting the roots is not that easy Then this method could also prove in very handy Is this fine? Any questions? All right, let's take another one question is question is proves that 2n factorial can be written as 1 into 3 into 5 all the way till 2n minus 1 times 2 to the power n into n factorial prove that 2n factorial Can be written as 1 into 3 into 5 all the way till 2n minus 1 times 2 to the power n into n factorial Let's do this. I'll give you one minute. I'm sure you would have seen this question in your regular textbooks Just type done if you're done. Okay. No need to Give the explanation I'm sorry. Madhav. I missed your message. Could you repeat the method again quickly? Okay. So method there was If you know that your terms are consecutive you can break the right side term also as consecutive terms product That is what I was trying to talk about Okay, never mind. We'll get more examples in future also when you are solving questions Done This example is basically to tell you guys that 2n factorial is not 2 factorial n factorial. Okay, right as you can see it's a much complicated expression than Anybody would wrongly take it to be Okay, so how do you do it? So very simple 2n factorial. We can write it as 1 into 2 into 3 into 4 into 5 all the way Let's say 2n minus 2 2n minus 1 till 2n Okay, I'm not writing all the terms. Basically, I'm just writing the initial few and the last few Now what I'm going to do is I'm going to take the odd ones together I'm going to take the odd ones together under one umbrella I'm sorry Okay, and I'm going to take the even ones under one umbrella So six and then dot dot dot dot dot in this and of course this okay So two four six eight dot dot dot till 2n minus 2 into 2n So what I did was I first wrote down them as a continued product From one all the way till 2n that's how factorial is defined And then I took the odd ones under one umbrella. I've written them in white And I'm taking the even ones under one umbrella. I've written them in yellow Okay, the white ones. I'm not going to disturb. Let them Be as they are Okay, while the yellow ones what I'm going to do from each one of these terms I'm going to pull out a two Right each one of them can contribute to a factor of two So if I pull out a two from here two from here two from here two from here two from here two from here How many twos or how many uh multiples of twos Or how many times twos will come out It'll come out as n times, right? So it'll become two Into two into two into two n times so two to the far end While when I do that it will leave me with a one over here See if uh two loses a factor of two it will be left with a one four loses a factor of two it will be left with a two Six twos is a factor of two. It will be three like this It'll continue all the way till I reach here. Okay And this term here is clearly n factorial. This term is clearly n factorial And this is what I wanted to prove. So let us conclude it Okay, easy. Most of you got it. Well done Next question is Oh, sorry Could you explain again? Sure, whatever See, whatever what I did first was I wrote the product I wrote twin factorial as the product from one till two n Okay, as you can see I've not written all the terms because of course it'll be too lengthy for me to write all the terms Now what I did was all the odd ones I multiplied them together One three five seven nine eleven thirteen da da da da till two n minus one Okay, I multiplied them together and all the even ones which I'm showing with a yellow Arc over here. So two four six eight ten all the way till two n I multiplied them together So with the odd ones I did not do anything. I did not disturb them With the even ones I started pulling out a factor of two from each of the terms So I pulled out a two from here two from here two from here two from here two from here two from here So like that I would collect two n times so two into two into two n times will come that will be two to the power n Meanwhile, the terms which have lost the two Will become from two to one Four will become from two to two six will become a three eight will become a four two n minus two will become an n minus one two n will become an n Now one till n product is known to be n factorial So I replace it with n factorial This is what I wanted to prove Clear by book Clear clear Okay Now my question is if if one plus Summation r into r factorial r from one to 50 is k factorial Find k Okay question is clear one plus Summation r into r factorial The this term is separate by the way, let me Okay, let me put a bracket around it r into r factorial r from one to 50 So you're putting first as r 1 then r s 2 then r s 3 and all the way till 50 and you're summing that term Okay, this entire sum this entire sum Along with the one added over here gives you k factorial What is k? What is k? Remember summation is only applied to this term. Okay, this is the term which is undergoing summation Okay Basically, I'm asking you what is 1 plus 1 into 1 factorial 2 into 2 factorial 3 into 3 factorial Not 3.3. Okay. Don't don't get it wrong. You cannot find 3.3 and 1.1 factorial So it is multiplication symbol, which I've used All the way till you reach 50 into 50 factorial. This is some k factorial. What is this k? That is what I'm looking for anyone correct aditya very good Aditya has cracked it Anybody else? Okay, never mind even if you're not able to do it Let's try to let us try to look into this term R into r factorial. Okay Now what I will do is I will write this r as r plus 1 minus 1 Okay, and this r factorial. I will keep it as it is Now you must be wondering. Why am I doing that? It'll become very obvious in few seconds. Just wait and watch Now if you open the brackets in this way You would realize that you have actually got r plus 1 factorial minus r factorial Now this is a very big achievement for us Which we call in the language of summation as method of difference Now, officially I have not done the chapter series sequence progressions with you But when I do that chapter You would understand this more deeply more closely because this is a type of summation which we normally use Are very close to the heart of the famous mathematician Srinivasan Naamanujan. So he used this a lot and Sometimes he misused it a lot also Okay, so when you write it like this, it brings it brings in a lot of advantages So what is the benefit and what are the advantages that it brings? See If you start putting your value of r as let's say 1 Okay, you get 1 into 1 factorial as 2 factorial minus 1 factorial Correct. So put your r as 1 throughout this expression Okay, then put r as 2 Then you get 2 into 2 factorial equal to 3 factorial minus 2 factorial Then put r as 3 Into 3 factorial is equal to 4 factorial minus 3 factorial and this continues. Let's say I reach I'll write 49 first then I like 50. So if I write 49 I get 49 into 49 factorial is equal to 50 factorial minus 49 factorial And finally when I write the last term 50 this gives me this Okay Now you'll see the actual benefit of breaking it like this. Okay, when you add it When you let's say add all the terms to the left side and the right side of this equal to On the left side, you will realize you are actually writing r into r factorial from r equal to 1 till 50 On the left, you will get this expression, which is basically what we require over here As you can see we require the same expression over here Whereas on the right side, you will see that cancellations will happen. This will get cancelled with this This will get cancelled with this. Similarly, this term will get cancelled with some term which will come over here Though I have not written it. It's very obvious This will get cancelled with this. This will get cancelled with some term over here like that ultimately All the terms on the right side of the equal to would get cancelled except for two of them, which is this And this That means on my right hand side, I'll get 51 factorial minus one factorial Right minus one. Uh, one factorial is actually a one and let me bring it to the other side So if you bring it to the other side one plus summation r into r factorial From 1 to 50 will actually become 51 factorial So this k value is actually 51 Okay Not a easy question, especially when you have not done the chapter sequence summation, but I thought this is a very good problem I should talk about it Okay, please make a note of this The same question will now will uh, uh Be converted into some other format when we do permutations. So let the permutation concept come in I'll give you a similar question There also Okay, note it down any questions any concerns. So this was the key This was the key Can I move on to the next question? Can I move on to the next question? Okay Yeah, see uh Aditya, you will realize when I do the chapter of Sequence series progressions There are certain instances where your regular methods of summation won't work For example, your sigma method of summation Right that doesn't work in many of the cases in those type of cases We tried to apply this method which is called the method of difference In fact, in this case, it is a special case of method of difference which we call as the vn method vn method So we'll we'll talk about it as of now. We don't want to get into too much of details Okay, we'll we'll talk about it when the chapter comes. We'll see various aspects of vn method Okay, one related question. I would like to ask here summation of R factorial from r equal to 0 to 100 If you divide it by 15 And find the fractional part of it What is the value? This bracket denotes the fractional part Okay, I think question is understood If you add the factorials some zero factorial all the way to 100 factorial And the sum if you divide by 15 You find the fractional part of that What will it be? Uh, you got you got the question, right? So it'll give you some number which has got an integer part and a fraction part Let's say I'm taking us a hypothetical case. Let's say it gives you 50.24 So 0.24 is your answer Of course that it will not be as Small value as 50.24. It'll be a huge value But I'm only interested in the fraction part Okay, Pranav Okay, Aditya Anybody else? Uh, can you give it in the exact form? Uh, Shashwath Exact form means P by Q form. No, no, no use of calculator, please Let me also take your attendance Okay and run Cool. Okay, let's discuss this Okay, I want to change your answer to that. Okay, very good now see Fractional part means Fraction part means the non integral part Okay, if you look at this term zero factorial one factorial two factorial three factorial four factorial five factorial And so on let's say I go till hundred factorial Okay, uh Cauchy It it cannot be a fractional part fractional part is always a number between zero to one your answer is Well above one. Okay Give me your answer in zero to one form That means remove any integer that you can get from there. Okay. Now see you are basically trying to get this correct that means this number This number is the fractional part right of This guy correct In other words, if this number is to be written as i plus f where f is some fraction Right where f is some fraction between zero to one Then basically i'm looking for f right Now all of you please pay attention 15 is made up of three and five right Now at the moment a factorial gives you both three and five into it From that factorial onwards there will not be any kind of fractions appearing Now what i'm trying to say here is If I break it up like this, that means I individually divide every term by 15 Okay, let's say I start dividing every term by 15 Just few of them. I'll show you and then you'll get an idea about what i'm talking about Okay So what i'm going to what i'm trying to say is that The moment a factorial gives you both three and five in its factors or in its expression From that factorial onwards That number would be completely divisible by 15 Now if you see this is one by 15 This is also one by 15 This is two by 15. This is six by 15. This is 24 by 15 But if you see five factorial five factorial is 120 120 Basically when you're writing five factorial, you will have one two three four and five That means three and five will be present in your Five factorial So they will get cancelled with the denominator as a result you will get an integer from there Are you getting my point so a pure integer will come out from there Are you getting it similarly this guy will give you seven factorial, which is 720 by 15 Which is going to give you a 48 A pure integer will come out Are you getting my point so all these terms till you reach hundred factorial By 15 they will all give you some integer terms Right, that means these terms will not contribute to any kind of fractions The only ones which will contribute to fractions would be these first five ones These guys will contribute to fractions Correct, and if you sum them you'll see that you have actually got 34 by 15 Okay, and this will give you some integer. Let's say i dash Okay, but remember 34 by 15 you can still extract Two from there as an integer so you can write it as two plus four by 15 Right, so two and i dash will become your integer of that required expression And four by 15 would be your fractional part this would be your fraction part Right, so your answer is going to be four by 15 sure i'll explain it once again see See you're dividing by 15 right here every term you're dividing by 15 So those terms which after getting divided by 15 doesn't give you a remainder Will those terms contribute to a fraction part? You'll obviously know Correct, so five factorial including five factorial all the way till hundred factorial Every term has a factor of 15 into it. So if you divide five factorial by 15 you'll get an integer Six factorial by 15 you'll get an integer seven factorial by 15 you'll get an integer Still hundred factorial by 15 you'll get an integer so all these terms here will be integers So they will never contribute to a fractional part Correct Isn't it Now the first five terms Which add up to give you 34 by 15 The fractional part will be hidden in this term So 34 by 15 I have further broken down as two plus four by 15 And two plus i dash will become another integer So only four by 15 will be left out as a fractional part Fractional part means that will only contribute to the decimal notations in that expression So that decimal notation would be four by 15 That is what I was trying to explain Does it make sense shares especially does it make sense? I wrote it just to explain that there is an integer part and there is a fraction part See I'm just taking an example. Let's say your answer comes out to be 220 Point three six. Okay. I'm just taking an example Right. So this can be written as two two two three four plus point three six What do I need to have a We need this guy. This is the fraction part right if somebody asked what is the fraction part of this You'll get this correct. So what I did I separated the integer and I separated the fractions Now what I'm trying to say is that All these terms will contribute to the integer part And this guy will contribute partially to the integer part and partially to the fraction part So here I figured out that two was the integer part and four by 15 was the fraction part Got it. Okay Any questions please do us All right Now one of the very important applications of factorial and a lot of questions have been framed on this in j main exam is finding the exponent of a prime number let's say p in Factorial of a number Now, let me first explain the name of the topic. Many people don't understand the name of the topic What are the meaning of exponent? What are the meaning of exponent? Can I turn it off? Yeah, what are the meaning of exponent exponent means power? correct Power of a prime number in n factorial. So if I ask you a simple question In five factorial, what is the exponent of three? Okay, in five factorial Find the exponent of three exponent of three That means if I prime factorize five factorial That is two to the power something three to the power something five to the power something and whatnot Then what will be the power that will appear on three? That is what we are learning in this particular concept That is what we are learning in this particular concept Right, so if you prime factorize five factorial Okay, what is the exponent of three? How will you get it out? Let's try to you know solve this question Five factorial is one into two into three into four into five I can see that only this will contribute to a three. No other term can contribute to a three So only one three will come out In the prime factorization of five factorial. So you will say that exponent of three in five factorial is one most of you have answered it very very good By the way, the notation that I would be using for it is E p n factorial Okay My camera is coming in between Sorry for these scratches Okay So if I ask what is e Three five factorial your answer will be one. Okay This is for your prime number which you're looking for and this is the factorial of the number in which you are looking for the exponent And that is one. Okay Now these kind of expressions may not be used without defining it Especially j doesn't use any, uh, you can say, um, you can say, uh, unknown expression Okay, they will always define it for you now This question was fine What if there was a very big factorial? Let's say I said 50 factorial and I wanted the exponent of three there How will you find that out? Okay, so let us understand the theory with this example Let us say I want to find out Let's take another example I want to find out The exponent of three in 50 factorial. How will you find out? Now if you write 50 factorial I'm just writing few of the terms so that, you know, you are Convinced with what I'm writing da-da-da-da Let me write a few terms like 18. Da-da-da-da till let's say 27 54 like that. Okay. Okay now If I want to know what is the exponent of three in 50 factorial, I would like to find out Which terms here would contribute to three? In fact, there are some terms which will contribute to multiple number of threes Correct So let us first figure out those terms from where I can extract at least one three Which will be those terms you will say sir all those factors of three Will definitely contribute to a three Correct. So three will contribute Correct. So three will contribute six will contribute nine will contribute 12 will contribute 15 will contribute 18 will contribute 21 will contribute 24 will contribute da-da-da-da till your last time 48 will contribute Correct. So can I say all these terms three six nine 12 15 18 21 24 27 blah blah blah blah Till 48 They will definitely contribute One three each Correct. So tell me how many factors of Three is present from one to 50? Your answer will be 16 Isn't it? Yes or no? Basically you are actually giving me The gif of 50 by 3 see why I wrote it like this because I want to formalize things I want to develop a formula for it. Isn't it? So can I say these many Many threes will come out from factors of Three present in one to 50 terms Are you convinced with this? Can I put a tick mark over here? Are you happy with this explanation? Correct. So one one one three I can pull out so Three six nine 12 15 18 da-da-da-da till 48. Okay Now is this all you will say no Because there are certain terms which will contribute more than one threes for example nine 18 Correct Correct nine 18 27 These terms will contribute to more than one threes Isn't it? So can I say I can extract 50 by nine such threes These many threes From those terms which can contribute at least two threes Okay, so from factors of nine so they will end up giving me these many more threes Isn't it? See three can only give you one three six can only give you one three But nine can give you two threes When you already extracted In the first round Right, so think as if they're like plants from where you can pluck out threes So one fruit you already plucked from it one more you can pluck from it Are you getting my point? Similarly 18 can also contribute two threes Because when you when you took a three for the first time it was left with six six can give one more three to you Are you getting my point? So can I say all those factors of nine which I left They are eligible to contribute to one more three Correct But is this the end of our game? You'll say no There are certain terms which have the power to contribute at least three threes Like 27 is such term 27 if you take a three from it it will be nine Nine again you can take a three from it it will be three and then again you can take a three from it So three threes it can contribute. So can I say I can obtain Gif of 50 by 7 this is gif operation So the sorry 27 not So I can obtain these many threes These many threes From factors of 27 Now I don't think so you'll be able to extract any more threes So all you can extract are these numbers of threes So your final answer here would be The exponent of three in 50 factorial would be Gif of 50 by 3 which is basically 16 in number Gif of 50 by 9 which is 5 in number Gif of 50 by 27 which is I think 1 in number. So this is 16 plus 5 Uh plus 1 So answer is going to be 22 threes you can extract from the number 45 Have I done a mistake somewhere? I didn't understand 45 uh shashot Okay, okay Fine, so if you write 50 factorial Then the exponent of three that will come out in the prime factorization of 50 factorial will be 22 That means 50 factorial can give out 22 threes out of it leaving behind a integer Leaving behind a integer Integer non divisible by three Okay Is this understood? Now this is what we are going to scale it up and make a formula out of it So let us scale this up In general In general If you are asked what is the exponent of a prime number p in n factorial the formula goes like this Gif of n by p Plus Gif of n by p square Plus Gif of n by p cube Will go all the way Will go all the way Till Gif of n by p to the power s where where s is such a number Where p to the power s is slightly less than n But p to the power s plus one will become more than n Okay, in other words just think as if You have to go till that term which doesn't give you a zero. Okay. For example, uh in the previous question I would not divide 50 by 3 uh to the power of 4 because 3 to the power 4 will be 81 50 by 81 fraction part will be zero. Okay till you don't start getting a zero Okay, this formula is basically known as the lujan's formula lujan's lujan's formula Okay written as legendary but pronounced as lujan Okay, this is known as your lujan's formula. Is this clear? Okay, now let us take up some questions on the basis of this Guys one small thing I would like to uh highlight over here This formula only works when you are finding the exponent of a prime number Okay, this will not work if you are looking for exponent of a non-prime number. However, we can still find it out. So other ways Let's take this one Find the exponent of three in hundred factorial the factorial symbol is missing. I'll add it up Let me add it up Yeah Find the exponent of three in hundred factorial Please put your answer on the chat box easy question Very good. So you have to find e 300 factorial. So as per the lujan's formula, it is 100 by 3 gif 100 by 3 square gif 100 by 3 cube gif 100 by 3 to the power 4 gif beyond it It doesn't make a sense to write because it will give me 000. Okay after this. So I don't want to write So 100 by 3 gif is 33 Now the trick here is most people sit and calculate each one of them. You don't have to do that Once you have calculated 33 you divide that by three and take gif So 33 by 3 gif will be 11 11 by 3 gif will be 3 3 by 3 gif will be 1 so your answer will be 44 plus 4 which is 48 Is this fine? Any questions any concerns can we move on to the next question? Find the exponent of 2 in The continued product from 20 all the way till 11 Find the exponent of 2 in the continued product from 20 all the way till 11 Okay, enough very good So this is very easy all you need to do is write the product of 11 12 all the way till 20 as 20 factorial By 10 factorial right, so this is like finding the exponent of 2 in 11 into 12 into 20 factorial is like finding the exponent of 2 in 20 factorial minus Exponent of 2 in 10 factorial isn't it so you see how many twos can you extract from 20 factorial? See how many twos you can extract from 10 factorial and since they have been divided by the exponent law The power of 2 will get subtracted So this will be gif of 20 by 2 gif of 20 by 4 that is 2 square gif of 20 by 8 Which is 2 cube gif of 20 by 16 which is 2 to the power 4 beyond that. I don't have to go minus This will be gif of 10 by 2 gif of 10 by 4 and so on Okay, not so on zero. I mean after that will be zero. So you don't have to write So this will give me a 10 plus 5 plus 2 plus 1 This will give me 5 plus 1 Right, so 5 plus 1 will get cancelled off So 5 plus 2 my bad my bad 5 plus 2 Oh, one more term can come. Why did I stop here? Sorry. Sorry. I was having an impression of different term. Yeah, sorry So 5 plus 2 plus 1 so I can add it to so I think 10 will be left off. So the answer is 10. Absolutely correct So those who answered with 10 Very very nice Okay, next is Let's take this question find the number of zeros at the end of 60 factorial Find the number of zeros at the end of 60 factorial that means if you write 60 factorial as a number What would be the number of zeros it will end with? What is the number of zeros it will end with? I'm not talking about zeros which are in between the non-zero numbers What will be the last trail of zeros that it will end with? Okay, panav Arithra nice Very good aditya Very good pratik Okay, mother Okay, now people who are getting six it is because you have misused the formula Okay, you have done gif of 60 by 10 you have done gif of 60 by 10 square and so on Of course, this will not give you anything. This will give you zero. This is a gross misuse of the formula What did I tell you the formula will work? That's why you don't some of you are not listening to me properly The formula will work only when you are finding the exponent of Of a prime number in factorial of something is 10 a prime number By no means Correct. So those who are throwing out the answer of six Surely you did not hear me out properly Okay So of course the idea is correct. That means you want to find out that if you write 60 factorial as you know something Okay, what would be the power of 10 that will come out? Okay, this is what you are looking out for right because the number of 10s you can pull out from 60 factorial That would decide how many zeros it will end with The idea is clear But the implementation is wrong Correct. So how else we can solve it? So how we can solve the exponent of 10 in 60 factorial now You just have to think it out not a very uh difficult concept first of all Think 10 is made up of which two prime numbers 10 is made up of two and five Correct That means one two and one five is required to make one 10 Right. So let me take you to some commerce chemistry terms One mole of two reacts with one mole of five to Produce one mole of 10 Correct. So think as if two and five are reactants and five 10 is a product Okay Now why i'm worried about this is because Now I will check how many twos are available to me in 60 factorial And I will check how many fives are available to me in 60 factorial Correct. Let us do that first Because that can be easily done because two and five are prime numbers. So I can use my lujan's formula there very easily So gif of 60 by 2 gif of 60 by 2 square gif of 60 by 2 cube gif of 60 by 2 to the power 4 gif of 60 by 2 to the power 5 gif of 60 by 2 to the power 6 I don't need to do actually because that will give me a zero Okay, so from here zero zero will come. Okay, so this will give me a 30 This will give me a 15 This will give me a seven. This will give me a three. This will give me a Uh, uh, uh one, okay One two three four five one two three four. Yeah, so let us calculate how many uh zeros how many twos will be generated So this will be 45 55 plus one 56 Is this fine Okay, now let me complete. There is something very interesting that I'll speak here Then e five If you find e five You'll end up getting 60 by five 60 by five square 60 by five cube Now here onwards I'll start getting zero. So I need not write 60 by five cube Okay, so this will give me 12 plus two which is 14 Now guys those who calculated e to six 60 you actually wasted your time So I purposely wasted it to show you why I wasted my time It is because If you know one two and one five is required to make one ten And I know number of twos which I can extract will always be more than number of fives I can extract Then I know that five will always be in a lesser quantity Five will be your limiting reagent in this case All right. Yes, sir So why do I waste my time looking for how many twos I will get when I know that one two and one five is required to make a ten and The number of fives will actually dictate how many tens I can make so why not I just find what is e five ten uh 60 factorial Why should I go for this? So if you're finding this you are just wasting your time Okay, so a smart problem solver. He'll just find e five of 60 factorial and that is 14 that means the power of 10 that you will get in the expansion of 60 factorial Will be 14 that means 60 factorial will end in 14 zeros Are you getting my point? So this is the first problem where basically we learned how to find exponent of a non prime number Okay indirectly so this question Indirectly helped us to understand that okay, so let me take a follow-up on this I hope this concept is clear Any explanation any concerns, please do highlight Let's take another one find exponent of exponent of or let me write it like this find e e uh, let's say 12 in 100 factorial Find e 12 in 100 factorial Yes. Yes. Yes. The chemistry analogy work works very well So first time you're seeing an application of chemistry in maths Up till now you were seeing application of maths in physics and chemistry But this is the first time chemistry is helping you to solve a mass problem Okay, very good Very good. Pooja, mother, Pranav, okay, Shashwat Aritra, Aditya, okay All right, let's look into this now 12 is not a prime number Right, but 12 is made up of two square into three to the power one Okay, very good Hariran Want to change your answer mother Okay, never mind change it no issues Now for this I need to see how many twos I can extract for 100 factorial and how many threes I can extract from 100 factorial Now read this as if One mole of 12 is produced by reacting two moles of two and one mole of three So I want to check how many moles of two is available to me and how many moles of threes available to me And accordingly from this Sociometry ratio, I'll be able to know which one of them is a limiting reagent for me Correct. So let us find out what is e2 of 100 factorial So this is uh gif of 100 by 2 gif of 100 by 2 square gif of 100 by 2 cube And so on I think I will go till 100 by 2 to the power 6 if I'm not mistaken Okay, so this will give me this will give me 50 This will give me 25 Then I'll get 12 then I'll get a 6 then I'll get a 3 and then I'll get a 1 So this is 75 87 87 and 10 97 twos Okay, so I have 97 moles of two with me Correct. Let us figure out what is e 300 factorial So this is going to be 100 by 3 100 by 3 square 100 by 3 cube 100 by 3 to the power 4 That's it. I don't have to go any further because I'll start getting 0 from there on This will give me 33 11 3 and 1 Correct, which is 44 plus 4 48 48 Okay, now Every one three reacts with two twos to generate one twelfth Correct. So you have 48 moles of three and you have 97 moles of two Correct. So can I say these 48 moles of three would require 96 moles of two to generate 48 twelfths? So thankfully e is my limiting reagent over here But let me tell you this may not always happen Because in this case the ratio is not 1 is to 1 it is 2 is to 1 had it been 3 is to 1 Then 2 would have become a limiting reagent Are you getting my point? Let's say I had a number 24 If I asked you The number 24 then it would have been the ratio 3 is to 1 Then you will have to check out whether which is the limiting reagent Accordingly decide how many product will be formed So in this case 3 is the limiting reagent your 3 is the limiting reagent limiting reagent If Gaurav sir sees these notes you will think that I've started teaching chemistry So there will be 48 12s produced in this so let me write it like this e1200 factorial will give you 48 Okay. Oh It happens these mistakes happen Fine. So I think we are pretty much done with the concept of factorials uh So we are now going to give this factorial