 Let's start. The heading is permutations involving identical objects or sometimes we'll say permutations with repetitions where the same letter appears more than once. And what we're going to do is a little experiment here. The permutations of the four different letters A, B, E, and F are now I've listed them for you. How many are there? Oh, no, don't count because we want to count without counting. How many letters are there grand total? Four, P, four. We're picking all four or you could go pick four, then pick three, then pick two, then pick one, but what's the short way to write four times three times two times one? Four factorial, no matter which approach you use, there's 24 ways to mix up four different letters and you can see we have six rows of four. What happens if two of the letters are the same? We're going to investigate this. What we're going to do is everywhere you see an F, turn it into an E like that. Do it for every single one, an example two. Turn all of the Fs into E's and so what we're saying is what if you have two E's in every word? So do this please. How many different ways are there to mix up A, B, E, E? Well, I want you to look really closely. Do you notice this column and this column actually has the exact same words? See it? This column and this column actually has the exact same words and this column and this column has the exact same words. In other words, I'm going to argue that we wouldn't count those or those or those because those are the same words. We don't want to count them twice. I'm going to argue that there are 12. How does this number compare with the answer from A? I'm going to write 24 divided by 2 and what we're looking for right now is a pattern and the first thing I'm thinking is oh maybe you just divide by the number of letters that repeat. That'd be nice. What happens if three letters are the same? What we're going to do is we're going to convert every single E and every single F to a B like that. Every single E and every single F to a B. Take about 30 seconds to do that real quick. Brett, do you have your graphing calculator here? Thank you. Vitalik, do you have your graphing calculator here? No, you aren't even here. Is he in this block here, isn't he? If we're mixing up A, B, B, B, now here's what I notice. It looks like the first row in every column is ABBB and then BABBB, in fact the second row is the same in every column and the third row is the same. In fact, every one of these columns is identical. I wouldn't bother recounting all of those because they're the same. You know how many different ways there are to mix up four letters if three of them are the same? How does this number compare with exercise one? Well, 24 divided by 6. I would really have liked it. Two letters repeating and a two there. Three letters repeating. I'd like to put a three there somehow. How can I write a three there but have it really be a six? And I'll pause and see if anybody can have the leap of brilliance because it will impress me. Is that not six? And you know what? I think that worked for the previous one because even though I didn't write it, isn't two factorial two times one two? I think Jasmine the rule is this, however many you normally have four factorial, four letters divided by how many are repeating factorial. However many letters you have grand total, four factorial divided by however many letters are repeating factorial. And I didn't notice that this was a factorial because of the fact that two factorial does work out to two. But three factorial, that's a six. That's a six. In fact, let's go really obvious. What if you had all a's? How many different ways can you mix up four a's? Only one. That would be four a's and four of them repeat. Oh yeah, that's a one. Let's generalize this. How many ways are there to mix this up? How many letters grand total? Five factorial over two factorial d's, calculator. Five math, left arrow, factorial divided by is two factorial the same as two? That I'm not going to waste time going to math, left arrow, down factorial. I'm going to just type two and I'll assume that you know that I did the calculation in my head because it's way less typing. And the answer is 60. Yes? What would the equation be, the expression be for this one? Five factorial over three factorial. Now what would I type into my calculator? Here as well, I would probably go five factorial. I would go second function, enter first of all. Yes? But instead of typing three factorial, I have ended up memorizing that I can do three factorial in my head. What's three factorial? Which is way faster than going math, math, blah, blah, blah, blah, 20. What's the answer going to be here? As an expression, what's the equation going to be, Brendan? No calculators, none, no calculators, put them down. Amrit, what is five factorial? What's four factorial? Wouldn't the four, three, two and the one all cancel on top and the bottom? In your head, what does this one simplify to? You don't have to do these in your head. I'm just going to tell you your thumbs will start to bleed if you do the easy ones in your head. Yeah, do the tough ones on your calculator. But I would do the easy ones in my head. And certainly, Jen, I will never type two factorial. That's two. I will never type three factorial at three. And I'll be honest, I also know that four factorial, which is four times three times two. I know that's 24. Those ones I just do in my head, it's way faster than trying to go math, blah, blah, and you'll see what I mean in just a bit. D, how many letters grand total? How many B's? How many C's? Now again, you can, if you want to, go five math, boom, factorial divided by how many terms, how many numbers are on the bottom? Two brackets. And you can go two math, boom, factorial, two math, boom, factorial, or I would go that's two times two. It's four. It's much faster to type. I don't care though. You can use your calculator. What will the expression be for E? Five factorial over. Can you see why the bottom doing that as 12 would be much faster? It's six times two boys and girls on the bottom. And that's what I would type on my calculator. I would go five math, left, factorial divided by 12. The reason I've only memorized two, three, and four is if they're giving you words, there's not many English words that have the same letter five times anyways. Two, three, and four will as a shortcut bail you out almost every single word in the English language. What was the answer Tim? So let's write this down algebraically. Here's the generalization. If you have n objects and factorial, but you have a of the first object and b of the second object and c of the third object, etc, etc, etc. So what's the difficulty here? Yeah, you got to look at the words carefully because I will give you some words with no letters that repeat over and over, but I'll also give you some that will. Oh, go Pogo. Dylan, how many letters grand total? It's going to be seven factorial all over. And then all I do is I walk along the word. How many O's for? How many G's to how many? Oh, I already did the O's. How many P's who cares? I already did the O's. I already did the G's. I already did the O's. Oh, I'm not going to type that into my calculator. I'm going to go seven math. I will do this, but I'm going to go divided by 24 times two because I know that four factorial is 24. That's way less typing. Is that okay? You guys okay with me doing the simple factorials in my head? Trust me, it's just it's a pain to type otherwise. 105. List them. No, I'm not going to ask you. The whole point of this unit is to count without counting. Statistician, two S's, three T's, two A's, three I's, S's I did, T's I did, I's I did, one C, I's I did, A's I did, one N. Now again, I am going to type 12 factorial in my calculator. What does this work out to? It's two times six times two times six. It's 12 times 12. I think that's, or even if you typed 12 times 12, that's still way faster than doing anything else. What's the answer? 12 factorial divided by 144? Read me the digits, David, one at a time, three, three, two, six, four, zero, zero. List them. No, the whole point of this is we don't want to list them. Okay? By the way, this can also be the, on your iPod, if you're making up playlists, but there's one song that you like so much that you like to have in your playlist more than once, right? In your teenagers, you're stupid that way, so I'm sure you do that. Don't turn the page. At the bottom of this page, we're going to play a little game. Okay? I need you to listen then. Supposing that you're signing up for a new website and the website generates a password and the way it generates the password is it takes the permutation of your first and last name as if it was one word. I would like you to write out your first and last name as one word. I'd like to find out who has the best first and last name and who has the loserest first and last name. So for me, mine is going to be one, two, three, four, five, six, two Ks, two Es, one L, one V, one I, one M, one D, one U, did the Es. Now, who's my money on in this class here? Jordan's away today. Oh, she picked a bad day to be away. Pardon me? Where's yours? But I said yesterday, you wanted to bring it every day. Give me your wallet, my friend. Here you go. All right. I'm going to hit pause on the video. Everybody get an answer? Now turn the page. Dina, what was yours? Oh, because it's those few extra letters. So you have lots of repetitions, but Amy's is both short and what do you have? Two As, two Ys, but still that, because that gives you factorials in the denominator that shrinks it down considerably, right? Example two, a true-false test has seven questions. How many answer keys are possible if three answers are T and four answers are F? And this is the neat part. I'm going to argue with you that what they want is, can you write this down, please? T, T, T, F, F, F, F. That's a word. How many letters in that word grand total? How many Ts in that word grand total? How many Fs in that word grand total? And that's why I said to you yesterday or the day before, we're looking at letters so often, Jen, because it's very easy for us to take any analogy and metaphorically turn it into a word. What do we get? 35, anyone else? 35? By the way, you need to practice how to do this on your calculator. Yes, Amrit. Now, Amrit, did you get 35? Say no. Amrit, my friend, this is why you need to practice this. Did you go seven math, back, factorial, divided by three math, back, factorial, four math, back, factorial. How does your calculator know that's in the denominator? How many numbers do you have? I was hoping, thank you, by the way, I was hoping somebody desperately would do this because how many numbers are in the bottom? What do I have to put here? Reckets. Or it's six times 24. You may be able to do that in the head 144. But brackets. Amrit, if this was multiple choice, do you think one of my answers would be 20,160? And you know what another one of my answers would be? If they did the Fs first and then they did the Ts next, that would also be one of my answers. Guarantee it. So don't do that. What do we say? 35, right? Because my calculator, because these aren't in brackets, it's only dividing by that number and it's multiplying by that number. So here, if it's in brackets, it doesn't make a difference. I'm trying to think of the most common errors that students will make. And I don't know whether they wrote the Fs first or the Ts first. Most of them would write the Ts first, but I got some weird students in this class that, you know, I can't figure out, right? Nothing. The grid did not appear. Do you guys have a grid on your screen? Bear with me for a moment. We don't have a screen either. Shut up. Do they have pathways? Okay. So here's what I want you to do. Sorry, I didn't realize that this had somehow vanished. I'm going to cheat. Insert table four by six. Whoops. Okay. Can you draw a little, come on. Can you draw a little four by six grid? Now, by the way, please notice to draw a four by six grid, you're going to have to draw five lines by seven lines to get four squares by six squares. And while you're doing that, I'm going to see if I can format the lines a bit and make them darker. Can I? Not in one note, probably. And in the top left corner, put location A, and in the bottom right corner, put location B. And the question asks, how many different paths can I take to get from A to B, as long as I'm not allowed to go backwards, as long as I have to always travel right or down, or right or down, or east or south. Matt says, almost seems infinite. I'm just waiting until everybody has the picture drawn. Oh, if you have travel backwards, it'd be infinite. Okay. I'm going to look at one path, the simplest path. I'm going to go east, east, east, south, south, south, south, south, south, south. Write that down. That's one way to get there. East, east, east, east, east, south, south, south, south, south, south. That's how many streets I would have to go on if this was a road, man. And here's my argument. Listen closely. I'm going to argue that no matter which path I take, I am going to have to go east four times somewhere along the way, maybe not on a road, but four times. And I am going to have to go south five times somewhere along the way, maybe not in a road. In fact, I'm going to argue six times, sorry, six times. I'm going to have to go south six times again, not necessarily all in a row, but somewhere I'm going to be going south six times. And I'm going to argue, that's a word. And I'm going to say when they're saying how many pathways can I take, I think what they're really saying, Dan, is how many ways can I mix up this word as long as it has four easts and six souths. I think it's 10 factorial all over four factorial, six factorial. Neat little application turns out maps are actually words, if you think about them the right way. Pathway problems. What's the answer? Brackets in the denominator, right, Amrit? And this one, I'd probably have to type in the whole thing, because I don't know what six factorial is off the top of my head. I think it's 720, but don't quote me on that. 10, you know what though? Math, second function, enter, second function, enter. Geez, second function, enter. And that'll work. One, insert a bracket. There's already a four there. Put a six there. Close the bracket off. That was faster than going through all those menus anyways. 210. It's going to be a pathway problem when you test. Probably a multiple choice. What are the most common mistakes? Kids count corners rather than streets. Instead of going 10 factorial, all over four factorial, six factorial, they'll go 11 factorial all over workbook, please. Can you turn on your workbooks, please, to page 389, page 389. And the first thing we're going to look at here are curveballs. Our permutations, it says, with restrictions. What it really means here is what if a certain letter has to be in a certain place. So, and how many ways can all the letters of the word oranges be arranged if, now double check the word oranges, any letter repeat? Nope. Okay. So I'm not worried about dividing, dividing, dividing. How many ways can all the letters of the word oranges be arranged if there are no further restrictions? One, two, three, four, five, six, that many ways. Which I think from last lesson, if I recall, was 5,040, wasn't it? I think. You start to memorize a few of these because you get sick of having to hunt for that silly menu after one. What if the first letter must be an N? One, two, three, four, five, six, seven. The first letter must be an M. How many choices do I have for an M? One. Now how many choices do I have? Six then, five then. You know what? Six factorial. I'm not going to fill in all my blanks because the blanks are purely to help me visualize what's going on. Six factorial is, is it 720? Is it? Okay. The weird one, because we've done like A and B already. The weird one is C. The vowels must be together in the order A, O, and E. In that order, side by side. This is where I think my little scrabble grab bag analogy works great because here's what I'm going to do. I'm going to imagine a weird scrabble tile that has O, A, E on it. That's one weird letter. And then I also have an R, an N, a G, and an S. If I force these three to be side by side in that order as one scrabble tile, how many letters are there in the grab bag? Not seven. How many? Five. The answer to this is going to be five factorial in any order. 120? No, yes. Yeah. Jasmine, what if they said they have to be together but they don't care about the order? So you can have O, A, E, A, O, E. Oh, you know what? How many letters are other on this tile if they didn't care about the order? Because there's that many ways to mix up the tile. But they didn't ask that. They sort of do an example two. An example two, they say, in how many arrangements of the letters of the word brains are the vowels together? So brains, yes. The vowels have to be together. So I'll call a weird scrabble tile AI, B, R, N, S. How many ways are there to pick these scrabble tiles? Okay, it's five factorial except they didn't say I had to have those in that order. I can mix those up too. I have another choice. It's not only mix up the tiles, it's also mix up these two letters. So since I have another choice, the fundamental counting principle says I'm going to multiply. How many letters are on this little scrabble tile? By the way, I know two factorial is two, but it'll be two times one. We really have five times four times three times two times one times. You got two ways to mix up the A and the I. And they're obvious. I, A, and AI. 240. Example three A. Just ask me what. Isn't it C? Which one are we doing? One, two, three, four, five tiles. Right? That counts as one big tile, a weird tile, but that's one weird letter. Two, three, four. There's five tiles in their grand total. Okay, trust me. You know what? Let's do three A and compare it. Yes, I didn't. For C, I did five factorial. Yeah, because here it didn't say that those letters had to be side by side. Here the scrabble tile, see, here the scrabble tile would look like this. A, O, R, A, N, G, E, S. We have to have an N at the front, so we got one way to pick that. How many letters are left? Six, then five, then four, then three, then two. Compare that to, let me get to three A with less chatter in the background, please. The word kitchen, but the K, C, and N have to be together, but not in that order necessarily. I'm still going to, Matt, visualize this scrabble tile with a K, a C, and an N as one big tile, and I also have an I, a T, an H, and an E. If these guys have to be together, they are one tile, how many tiles are there in my grab bag? Five factorial, but apparently I can mix these guys up. How many ways can I mix these guys up? Three factorial, because there's three letters on that tile. Hundred and twenty times six, seven thousand two hundred, no, seven hundred and twenty? Okay, yeah, whatever. We're not using long words, so a lot of our answers will kind of overlap a little bit. Do you want me to pull up Mississippi? I can, I prefer not to. Skip beat. Example four. And I apologize, folks, because of the short class, since it's 20 minutes short, we're going to have to go in right to the tone, but I want to, I run, because we're looking at the weird ones. This is where we really want to start to wrap our brain around this. Example four. How many ways can three girls and four boys be arranged if we have to go boy girl, boy girl, boy girl, or girl boy girl, boy girl, boy. First of all, if I have to alternate genders and I have three girls and four boys, what gender must I start with? In fact, I'm going to argue that one, two, three, four, five, six, seven. How many choices do I have for the first boy? Four. How many choices do I have for the next boy? Three, then, two, then, one. How many choices do I have for the first girl? Then, two, then. Now I wouldn't type that into my calculator. Can you see it was really four factorial, three factorial, and that's probably faster to type. Oh, wait a minute. I know four factorial. That's 24. And three factorial is six. 144. It can wait. Where is that math? Oh, on your iPod. Yeah, it's there. Yep. Sorry, I thought you were texting and I was going to lovingly freak. Six actors and eight actresses are available for a play with four male and three female roles. How many different cast lists are possible? How many males? Six grand total. How many roles are available? One, two, three, four. How many choices for the first male role? Then, then, then. Now the faster way to do that would have been to go from six males, permutate four of them. But I told you last day, if I'm not sure, I fall back on the fundamental counting principle and drawing the blanks. But now that I drew that, I went, oh, I could have done that. And we have three female roles, eight of them, seven of them, six of them, or I could go times eight p three. Either of those will give me the same answer. Which one's faster to type? Probably the bottom one. One, two, zero, nine, six, zero. And again, the numbers get very big, very fast. Okay. We're going to skip six. Then it walks through permutations with repetitions, similarly to the way that we did in our lesson. And all we're going to do is we're going to scroll down on page 391. I'll put a highlight around it. Here is the equation for permutations with repetitions. This is not on your formula sheet. This, you have to know. Although I don't think it's a difficult one to remember. In fact, Madison, I would almost argue it made sense. Almost like you're like, oh, I, it wasn't expecting the factorial part, but the two and the three and the four appearing made sense. And oh, factorial. Actually, now that does make sense, actually. Okay. Are you okay with doing, if we could, the word Vancouver and mathematical? I think you folks got the hang of that, and I've been talking too long. So I'm going to go straight to saying, here are the questions that you can try. One A, one B, one C. Yeah, D, I could have done all of number one. Two A and B, three A and B and E. I'll sign D2. Skip C. Six, eight, ten. Someone wants me to do Saskatoon. It's either going to be 12 or 14. Let me see here. They look long, but when you get the hang of it, they start to go pretty good. In fact, I would argue though, the first half of the homework that I gave you is trickier. It's with restrictions, you have to really think things through.