 Okay, so to summarize what I just said so far over here you can write down permutation We say The order matters and that's what we've been doing up until now. We counted it a b as different from b a for a combination We don't care. So let's try and give you a good example of the difference between the two If five sprinters compete in a race How many different ways can the metals for first second and third place be awarded? Well first place second place third place How many choices do I have for the first place metal? five How many choices do I have for the second place metal? Four, how many choices do I have for the third place metal? You'll notice when I wasn't sure I fell back on the fundamental counting principle But now that I see five four three. I recognize that's actually five P Three which might be a faster way to do it on my calc. Well, you know what I'm not gonna do this on my calculator It's 20 times three David. What's the answer? 60 at the Olympics does the order that you finish matter is a gold different from a silver different from a bronze? absolutely, this is an example of a permutation of Five objects taken three at a time However, what if you're at the Olympic trials now at the Olympic trials the top three advanced to the Olympics? You don't care whether you finish first or second or third. You just want to make sure you're part of the group That would be an example of a situation where the order did not matter in fact if five sprinters compete in a race And the fastest three qualify for the relay team. How many different relay teams can be formed? You can visualize the five sprinters below Three will qualify two will not you can kind of think of this as saying I'll have three yeses and two no's Why that looks like a word How many letters long is that word grand total? How many why's? How many ends what's the answer? 120 divided by 6 times 2 120 divided by 12. I think the answer is 10, but double check me is it 10? And I think I mentioned to you already last day the reason we start with words is if we're clever We can turn a lot of questions into words Does the order of the three matter here? No This is an example of a combination of Five objects taken three at a time and here is really the difference Look up That's the permutation That's the combination Permutation Right because if I went 5p3 or five times four times That is five times four times three right there because the two of them will cancel permutation combination and That's how we're going to have to tweak our factorial formula for our formula sheet as well so an unordered arrangement of distinct objects is called a combination and The number of combinations of n distinct objects taken are at a time is We write it as n see are For choose or combination And it's n factorial all over n minus r factorial now, that's the permutation part of the equation That's this part What's the three? What's the three? What's the three? Are factorial in the front on the bottom? It's n factorial all over our factorial Bracket and minus r factorial and if you look at your formula sheet or if you look on the inside cover of your workbook That also is on your formula sheet Yes, in fact, it should be right next to the permutation equation. Yes Okay Really the trickier part is trying to determine is this a choose combination or is this a pick permutation? But I've tried to do with my little model up here with my individual ease smacking battalion around I tried to give you a visualization so How many different committees of three people can be formed from seven people if We're just forming a committee. Have they mentioned anything here in this sentence like it might matter when you get picked This is going to be seven choose three and Now I have no idea what the answers are you are going to have to go to your calculator Let's go find where it is Seven math left arrow. Oh, there's the choose option number three three Seven choose three is 35 Now compare question a with question B How many different committees of three people can be formed from seven people if the first person Serves as the chairperson the second as the treasurer and the third as the secretary Does this suggest that the order makes a difference? Yeah, if you're picked first, what are you? Press chairperson if you're picked second, what are you? Treasurer you pick third to the secretary. So here I'm either going to go seven permutate three or I could fall back on my seven six five In some ways, I have to be honest make it I almost prefer this because I can usually do these in my head It's 30 times seven 210 yes someone double-check, but it's 210 by the way Are there more permutations or are there more combinations? There's also kind of a built-in error check if you got a combinations bigger than a permutation probably you've messed up some see I Like see I like see I like see I like see I like see I like see If the group of seven people consists of three males and four females. So now we've got subgroups How many different committees of three people can be formed if you want one male and two females? Looks tough, but surprisingly easy if I give you one more tool So what are the tools I've given you so far this year? I've given you the drawing a blank and the scrabble grab bag for permutations The best tool I can give you for combinations is to draw a bucket put your pencil down for a second and Watch if we draw this properly how easily the equation is going to come out So over here in the margin I'm going to draw a bucket and I notice there are two groups males and Females don't write this down yet. Watch me draw it and then write it yourself in that way you see it twice How many males grand total before we start? three How many females grand total before we start? Four how many males on the committee? I draw a little arrow. I put a one right there. How many females? To so I set it up like that takes one second to draw and here's the equation from three males choose one and From four meal make females choose to see what pops out of there. That's not bad That's what I stole. I think I mentioned at the very beginning of this There was a prof from north band who did this and instructing us if you ever go to Capilano College and you have Ted Bentley He's brilliant. He's the one of the few math university profs But I've seen that explains things at a high school level and explains it well when he showed me that I was Oh Four aces choose to four kings choose to That's four and four is eight fifty-two minus eight Forty-four other cards choose one. There's a full house with two kings and two aces We'll get there, but it's very easy to set up card questions then. What's the answer? 18 is it do you remember when we did some factorials? I said out of laziness Because it's a lot of typing. It was worth memorizing two factorial three factorial and fourth factorial Can you find for me five choose zero? What's that? What's five choose zero? Can you go second function enter change the five to a six? What six choose zero? I'm going to memorize that anything choose zero is one Elizabeth. Can you give me some help? I'm going to memorize that anything choose zero is one and that because a lot of the time You're going to have a choose zero in your equation, but you're not going to type it Can you find what's five choose one five? What's six choose one go second function enter change the five to a six? What seven choose one? Okay, I'm going to memorize that n choose one is just plain n because that's way faster to type Do you have to Nicole? Nope Will you want to? After your fingers start bleeding. Yes D D Read D very carefully to yourself, please and What's the difference between C and D? There's one key phrase that makes it a completely different question At least and it's so important Jen. I'm going to underline it in red The fancy term for this we say that this is looking at different cases and To solve these what we have to do is we have to list each case by the way. Did you hear what I said back here? I said one male from three and Two females from four. What does the word and mean in combinatorics? multiply okay, if We have at least one male. What are the possibilities we can have one male or Or Two males or Three males or Okay, I can't have fork only three guys. Okay. Those are the three possibilities. You know what or means in combinatorics Now one male we already did three choose one four choose two or Can you look at this bucket and modify it for two males without drawing it from three males choose how many? to so from for me and females choose how many one or Can you look at this bucket without redrawing it and modify it for three males from three males choose? three and From four females choose By the way, are you starting to see why if you were typing this you put a three there a four there and a one there? Actually you wouldn't you know what you would do look up look up look up. I Would go one term at a time. I would go three math factorial combinatorics choose one times four math Left arrow choose two and I would Write down that Plus and then what I would do is just go second function enter change the one to a two Change the two to a one and I would go 12 Plus and then I would go second function enter Change that to a three Change that to a zero by the way. What is four choose zero one? What do you get oh? Not only is four choose zero one you know what four choose four is one So anything choose zero or anything choose itself is one I actually knew this one was going to work out to one the answer is 18 plus 12 plus one in your head, please 31 Yes, that's how we're going to do cases and I like this question. I like this question I like this question to classic provincial exam written question on this unit is a committee question They used to do card questions But apparently a religious group complained and felt that that was inappropriate I guess there are some religions that have issues with playing cards So the odds are very good no pun intended with cards and odds But the odds are very good. It won't be a card question on your provincial exam But I'm gonna do I like it if I offend you so be it example to Example to says evaluate that Can my calculator go 100 factorial? But I can't help noticing does the bottom add to 100 This is actually 100 choose three or It's also 100 choose 97. What's the answer? What is it? Sorry? What's the answer? Read it to me please Brenda one digit at a time one 161,700 By the way, why do both of these have the same answer and you'll find 100 choose for is the same as 100 choose 96 or 10 choose 3 is the same as 10 choose 7. Why are they the same answer easy? Can the five of you stand up, please? If I form a group of two say there if I form a group of two am I not automatically Three in other words, you know what five to two is the same as five I can't form a group of four without forming a group of one. So if you notice Five choose one will be the same as five choose four or ten choose six will be the same as ten choose Four or twenty-five choose eight Okay, that can also be a time saver if you're writing out case questions Be Solve this okay, if this was multiple choice Nick, you know what I would do I would quickly try all four answers But it's probably gonna be written I told you there's gonna be some kind of factorial question to solve on your test on your formulas sheet What is n choose two if we plug in n for n and two for r? I think we get this n factorial all over two factorial n minus 2 factorial Is that correct did I plug it in correctly by the way, what is two factorial two? So on my next line, I'm just gonna evaluate this. It's not that the exclamation mark vanished It's that I went to factorial is two times what it's two and what I'm gonna do I'm gonna get all my numbers to one side So I'm gonna take this to and I'm going to move it over here by multiplying by two I'm gonna say this is the same as n factorial all over n minus 2 factorial equals 42 Okay, so I said two factorial is two and I multiplied by two to get the ends on the one side Like we did with logs to get all your x's to one side here get all your ends to one side now what? Which one's bigger top or bottom the top? Let's expand it This is going to be n times n minus 1 times n minus 2. I'm gonna stop there factorial All over n minus 2 factorial. Why am I gonna stop there? Can I cancel it is factored ha ha What kind of an equation is this? It's a quadratic You know how I know there's going to be an n squared in fact when I go chunk chunk I'm going to get n squared minus n Can I do this in one step minus 42 equals zero did I lose you there or is that okay? Because I'm running out of room here Oh sad was way ahead of a Sabbath factors into what? minus 7 plus 6 What are my roots n equals 7 negative 6 which one's extraneous? You can't have negative 6 objects Lovely question there Let's play cards So we're not calculating odds yet all we're doing is calculating how many a Standard deck and I need to explain this because some of you may not be familiar with playing cards a Standard deck has four suits clubs diamonds hearts and spades and 13 cards in each suit so how many cards are there grand total? Let's make a little note here 52 cards how many black cards are there? More specific how many black cards are there? You know mr. Dewick you should really write that in black 26 black and 26 red and Then the other one they occasionally ask is How many face cards are there now? What's a face card? Okay, a face card is a jack-queener king one of those three so how many are there? Well face cards All right, how many different five card hands can be formed? Cards are combinations. It's from 52 cards choose five So we're not playing seven card Texas hold them right now We're going five cards and the reason we're going five cards is crunch that the numbers get big Seven would be worse with five cards. How many five card hands are there? What is 52 choose five? Matt read me the digits one at a time just about is that right just about 2.6 million different five card hands So whoever was saying oh, this is a good method for counting cards. Well, I Guess if you're a method mathematical savant and can do huge Map in your head short be How many different five card hands can be formed that consist of all hearts? I'm gonna argue that this is the same as the guys and girls question except here We're gonna have hearts and non-hearts. I'm gonna draw a bucket and I'm gonna have hearts and I usually just write others how many hearts are there in the deck? Not oof. How many hearts are there in the deck easy? 13 how many others are left over in the deck then do the arithmetic? 39 how many hearts do I want to choose? Five how many others do I want to choose? Zero can you see what my equations going to be? 13 choose five and 39 choose zero by the way, what is anything choose zero? So I'm not gonna type that on my calculator, but I almost always write it for the symmetry of the bucket so I can see what's going on 13 choose 1287 put your pencils down. We would call that a heart flush Not a bad hand How many different suits are there? So if you want to know the number of flushes grand total times by four if you want to know the odds of getting a flush divide that by two five nine eight nine six zero and that's the odds of getting a flush Okay, that's how they calculate them the only problem with this is there is a special flush That's even better than a regular flush a straight flush So really what we should have done is figured out how many straight flushes there were Subtracted those from the 1287 times four and figured out the odds of getting a straight flush and then figure out the odds of getting a normal But anyways, this is how they do it. How many different five see How many different five card hands can be formed that consist of all face cards bucket face cards and and I usually abbreviate others as OTH because I get tired of writing out the word others How many face cards are in the deck? Well How many others are in the deck to the arithmetic? 40 how many face cards do we want? It says all so five How many others do we want then? Zero what's the equation? 12 choose five times 40 choose zero Which is really just 12 choose five Doug what'd you get? I know pick up your calculator. You want to try this? I Don't know where you are, but it's not here. I'm help me out. I know you're tired. Ah, no nothing Jack Kelvin what is 12 choose five? I Heard two different answers. Sorry what oh 793 I'm sorry 792 by the way, what if instead of all what if I'd said four face cards and one other What would the equation be? 12 choose 440 choose one this bucket very flexible, which is when I saw that I was like oh Wish my profit taught me that in college because that would have made life easier. Sorry How many different five card hands can be formed three hearts two spades? Okay This bucket is going to have three sections hearts spades and Others because some cards are neither hearts nor spades How many spades are there in the deck? 13 how many hearts are there in the deck 13? How many others are there in the deck? 26 How many hearts do I want? three how many spades To how many others none can you see the equation? 13 choose three and that's three hearts 13 choose two three hearts and three spades and No others that was Jordan's ringtone wasn't it? Oh, no, okay It's okay. Are you just for curiosity who here is a machine gun sneezer? So who here sneeze multi-seize multiples they're out there my little brother is like always six to eight in a row I've always wondered is that genetic or is that learned behavior? Anyway, it's just wondering what do you get David read me the digits two two Three zero eight is that right? Okay, and Again, you've heard me say this already, but a bear is repeated. There's no way you could count those by hand We're counting without counting E and F are almost identical questions. What's the difference between F and E? At least so F is going to be cases Let's draw our bucket. I'll draw my bucket over here this time So F we're talking about hearts. Do they mention any other groups? Nope, then we'll call the rest others How many hearts are there? 13 how many others are there? 39 and we want exactly three hearts Oh, so how many others will we have to pick by default then to? so this is going to be 13 choose three and 39 choose two you get 211 926 Yep 211,926 now as soon as they give me cases Because Jen I've done these sloppily wrong so often I have a couple of Mr. Do it being hyper paranoid first of all I Underline the case because sometimes I've read it wrong sometimes I've read it as at most Which is a completely different question from at least and then I'm gonna list the cases so at least three hearts means What's the minimum number of hearts I can have? Three and I draw a line underneath that and then I write the word or what's the next possible case? At least three hearts. What else could qualify for at least three hearts? four or Five Can I have six? Nope. We're only picking five cards Now we already have three. It's 13 choose three and 39 choose two, but I find sad by setting it up nice and wide and spaced like this I can usually just drop the plus sign and I can look at this in my bucket and figure out what this is going to be Four hearts. What would that be? What choose what times what choose what? 13 choose four and 39 choose one Or Five hearts, what would that be? 13 choose five and 39 choose zero Then Elizabeth to minimize my typing now. I've already done this first one. It's Two one one nine two six This is where I don't do this all at once I go case by case because if I just go second function enter I can change the three to a four and I can change the two to a one way less typing 27 885 and I go second function enter again, and this is going to be 13 choose five and 39 choose zero and I get 12 87 Which is the number that we had earlier for the number of ways to get a heart flush And what am I going to do with these three numbers add them up plus? 27 885 plus two one Mr.. Two one one nine two six I find that's about the least amount of typing and still have good bang for my bucket 241 oh nine eight turn on your workbook, please to page 397 page 397 Doug is back. Whoo-hoo so Combinations we already talked about so turn to page 398. They're walking you through the combination derivation Which is lovely, but we've already done that so page 399 what we are going to do is with our highlighter or a pen or something like that say hey Here's the equation Now in North America, we write NCR like this in Europe Do I have my any German exchange students in this class? They write NCR as n over r in giant brackets So I've showed you that just in case when you're in university you have a European prof or whatever both are accepted as notations But it's n factorial over r factorial n minus r factorial so lotto 649 You choose six numbers From 49 how many numbers would you have to make up to guarantee a winning ticket? From 49 numbers choose six and what does that mean? It means if the prize is more than 14 million dollars It's worth doing Or is it what's the risk you're taking if you buy one ticket of every number if you invest 14 million dollars What's the risk you're taking? Yeah Someone else also picks that number you got to split the prizes. That's a pretty big big crunch Okay, if the prize is around 50 million But the problem is the bigger the prize what happens around Canada More tickets are sold the odds of someone picking your same number Increases this is why no one does this. It's not mathematically feasible One three nine eight three eight one six one three nine eight three eight one six But basically the odds of winning lotto 649 are about one and 14 million you kind of wondered right, okay? Example three says we're gonna have an athletic council We're gonna have seven members I'm gonna underline the number seven because they wrote it out and I might miss that if I'm in a rush map We have nine males six females The subcommittee has to have three females so that also means we have to have four males if there's seven members Skip a skip B. How many ways can I have the subcommittee bucket? males females How many males are there grand total? Nine how many females grand total six Four of them have to be on three of them have to be on nine choose four and Six choose three What's the answer know what I'm gonna go second function enter second function enter second function enter hey that'll work delete nine choose four delete Six choose three that's still less typing I think 25 20 is it 25 20 Okay, all right mini curveball number one D In how many ways can the subcommittee be chosen if Bruce the football coach must be included How many ways are there to pick Bruce? one and Okay, how many males are now left in the bucket? Let's assume Bruce is a male. Thank you so How many males are now left in the bucket eight? Choose now we've already put one male on the committee. How many males are we now going to choose at random on the committee? three and How many females six choose still gonna have three females? So if they give you one restriction, it's actually pretty easy pull it out do it first and then deal with the rest of your bucket What's the answer? Oh, you know what I'm just gonna go like this make that a three Make that an eight 1120 turn the page Cards we've already done so as much as I'd love to do number four as a nerd we're gonna skip it and We're actually going to turn to page 400 and three Questions with at most at least etc So on your own right now try one see I'm gonna freeze the screen See if you can do this one. I think those are my cases Is that right? At least one female means one or two or three or four or five because it does say five council members right there All right Oops, I missed one. That was silly. Oh, I see what I did that was silly Did you all get an expression that looks like this You get five choose four six choose one plus five choose three six choose two plus five choose two six choose three plus five choose one Six choose four plus five choose zero six choose five Shortcuts look up. What's anything choose zero? What's six choose five I'll give you a hint. It's the same as six choose one This whole thing is gonna work out to six What's six choose one Six it's also the same as six choose five. What's six choose one? Six what's five choose four? That's also five because it's the same as five choose one Remember when I separated the groups this whole thing works out to 30 it's a little less typing if you spot stuff like that Would I do that on a test? Probably not Jasmine. I'd type it on a test when I try some shortcuts in my homework. Yeah All right, let's type this one five. Oh, you know what mr. Duk Just go second function enter and change the equation that you have five choose three. Oh Jesus almost is already there 150 plus Make that a two make that a three 200 Plus make that a one Make that up for 75 okay 30 plus 150 plus 200 plus 75 plus six I get 461 anybody else. Yeah, okay Almost done Dina What about cards if I said at least one red card and we're picking five cards What are the cases for at least one red card? one red card or two or three or or or Five that's a lot of work. I am gonna do B at most two Kings. Okay Here's my bucket Kings and others How many Kings are there in the deck? for How many others? 48 at most two Kings means what cases? two or one or Okay two or one or Zero sorry. I had no shortcut for cases. It's just you got to do them each individually Two car two Kings would look like this two and three There's my five cards. So two would be four choose two 48 choose three or let's see if we can fill in the rest without redrawing the bucket or by the way look up Easy way to do this is to now go cross that out cross that out One and four now you've got your new equation Four choose one 48 choose three or cross that out zero cross that out five. Oh four choose zero 48 choose five What's the answer 103 776 or I made a typo here. This should be a four in my writing shouldn't it make that a four Make that a one 778 320 holy smokes or make that a five Make that a none one seven one twenty three oh four one seven one twenty three oh Plus 778 three two zero Plus one oh three seven seven six Two five nine four four zero zero What if they said? Three or more Kings That would mean three or four or five. There would be a shortcut. I would say well, there's sorry. I would go 52 Choose five That's how many different Hands there are Here's how many have two or less That's how many have three or more That's a short way if they ask you a part B. You can use the compliment. Oh It is three or four Kings a good hand That means three of a kind or four of a kind There shouldn't be very many of those it should be a tough one to get all right Homework turn back, please to page three sorry page 400 and Well page 400 number one number two Pete's perfect pizza company. Okay, so number one number two because it's basketball seven 11 then we're going to skip ahead to Page 406. Yes, you are correct About Number two number three number six movies sure We'll pause there tomorrow's lesson is much much shorter