 There's the magic beep. So we're going over the homework. I already did number two. Number seven, how many even four-digit numerals have no work? Well, they're talking about four digits. So Troy, the first thing I'm going to do is one, two, three, four. If it's even, what makes a number even? It ends in a zero or a two, or a four, or a six, or an eight. By the way, most of us normally rattle it off in order as opposed to just kind of randomly scattering it like, see, that's okay. Oh, but if I'm also doing four digits, there's a restriction on the first digit. What can't the first digit be? Here's the problem. Do you notice that the number zero appears in two restrictions, in both restrictions? This is why this is a tough question. We're going to have to do this as two separate events. We're going to have to look at all of the even numbers that end in a zero. We're going to have to take care of that one restriction first. Then we'll look at all of the even numbers that end in a two, four, six, or eight, and don't begin with a zero. I'm going to, to help me kind of visualize, I'm going to draw my little Scrabble Grab Bag right here, because it really does help me visualize. One, two, three, four, five, six, seven, eight, nine. Sorry, I didn't just scatter them around, but I did instant the bag for you, so they're not in nice order. So I have to look at two cases. I have to start this question over, and I have to say ends in zero, one, two, three, four. Why do I have to separate them? Because that zero as a restriction appears in more than one restriction. We don't have the math skills to do them both at once. We have to then look at each individual case. So if it ends in a zero, how many ways are there to pick a zero? One. So I picked a zero. How many choices do I have for the first digit? Nine, then, then, okay. Or it ends in a two, or a four, or a six, or an eight. How many choices do I have here? Four. So let's suppose I put an eight on the end. How many choices do I have here? Not nine. Why eight? Can't have a zero. So I have eight choices here, because I can't have zero. So let's suppose I put a, oh heck, five there. How many choices do I have now? Well now there's no more restrictions, so now the number of choices is just how many do I have in my Scrabble Grab Bag? Eight again. And then seven. Total answer is going to be that plus that. Okay, that's one way of doing it. The other way you could do it, Troy, is you could find out the total number of four digit numbers, which would be nine times nine times eight times seven. Find out the total number of odd numbers, because odd is always easier to find, because odd, you'd have a one, three, five, seven, or nine here, and a zero here. Zero doesn't appear in both. I could just go to town, and then I could subtract whatever's left as even. Does that make sense? This is a tricky question. Don't kid yourself. You really got to, but do you see why I had to panic when I drew out my four blanks and I listed my restrictions? Uh-oh, it's appearing in both. I can't do it when it's appearing in both. I have to just look at them separately as separate cases. That should be it, by the way. I don't know the answer, but somebody try to be right. Yes? Any others? Well, A or B? Both? No restrictions, four letters. One, two, three, four. I'm going to do my little scrabble grab bag analogy here. P-R-O-D-U-C-T. How many choices do I have for the first one? No restrictions. Oh, does it say no repetition? It doesn't say no repetition. Did you go seven times seven times seven times seven, and that's wrong? Okay, I think they want no repetitions, and I think their hint for that is distinguishable four-letter arrangements. I think what they're saying is you can't use the same letter twice. Actually, no. You know what? They're using the word arrangement, which today I'm going to tell you means you're mixing and matching, but you can't pick stuff twice. You're moving the scrabble tiles around, but you don't have two-letter P's. So once you place it, you're just mixing and shuffling. So it's going to be seven times six times five times four. The more interesting one is B. If the word has to begin with P-R, here's what I would do. I would redraw my scrabble bag, and I would have an O, a D, a U, a C, and a T in it. But then I would also have one weird scrabble tile that had a P-R on it. That's how I would treat this, and I would say, okay, how many ways do I have to put the P-R? Well, it's going to be one great big scrabble tile that's made up of two letters, but I only have one way to put it there. And once I pick those two guys, then I have one, two, three, four, five times four. I think the answer is 20 to B. And that's already approaching harder than they'll ask on the provincial. They keep it pretty straightforward on the provincial, and part of it is it's very easy, very quickly to get very difficult with very easy-looking questions. Any others? Dylan, isn't this a nice change from identities and trig? Fun is the wrong word, but to me, this is way more intuitively interesting and intriguing. Go on once, go on twice, no more. So we're going to do lesson two, which is going to be a handout. Now, if you want to, you can turn in your workbooks to page 383, which is the next lesson. But I'm going to do the lesson from the handout that I'm about to give you. Lesson two, and you want to get your graphing calculators out, and you need your graphing calculators, because I'm going to be showing you where some very obscure functions are, and you're going to be using them all the time. That means if you don't have a graphing calculator here, first of all, you do want to bring it for this unit. We're not going to be using the graphing portion, we're going to be using the stats portion of it. But if you don't have one here, mess up, be embarrassed, I'll make fun of you, but you want one. Holy smokes, everybody have a graphing calculator? He's got more. I'll loan you this in class today, later on we'll write down the number. No, you're going to actually borrow one from me, I'm going to write down the number until you get yours back. Lesson two, permutations involving different objects. And you know what, underline or highlight the word permutation. This is going to be one of the key words in this unit. We're going to be looking at two main ideas for the rest of this unit. Something called permutations and something called combinations, and they're different things. Investigate, it says. Two letters, A and B, can be written in two different orders, A, B and B, A. These are the permutations of A and B. Permutations, Justine asks, how many ways can you mix stuff up? List all of the permutations of three letters, A, B, C. I would probably do it systematically, I would go well. You can go A, B, C and A, C, B. Those are all the ones that begin with the letter A. What other letter could you begin with? Okay, so you could go B, A, C and B, C, A. Those are all the ones that begin with the letter B. What other letter could you begin with? This is called systematically listing things. C, A, B and C, B, A. I think because we've done it systematically, I'm pretty confident we got them all. How many permutations are there? Six. B says, list all the permutations of four letters, A, B, C, D. No, not gonna. I could. A, B, C, D, A, B, D, C, A. I'm not interested in the list. I'm interested in how many? How many are there? Well, how many letters are there grand total? Four. Let's draw four blanks. How many choices do I have in my Scrabble bag for the first letter? Okay, I picked one of them. How many choices do I have left now for the next letter? How many choices do I have left now for the next letter? And then how many? I'm going to argue that I think there's 24. Now, let's see. Let's go back and test it with our three-letter case. See if it works there because three letters, we listed them all. How many choices would I have for the first letter? Three. Then? Then? Does that give us six? Ah. That's way faster than listing them. So, C says, predict the number of permutations of five letters. One, two, three, four, five. How many choices do I have in my Scrabble bag for my first letter? Then? Then? Then? Then? 20 times six, 120 for those of you who reach for your calculators, please. I'm not mentioning any names whatsoever here. We always start with letters because what we're going to do whenever they give us a nasty question, we're going to try and turn it into a word. We're going to try and somehow transpose it into letters. For example, instead of arranging letters, we can arrange objects if they're all different. How many different ways can five people be arranged in a line? Well, I could call the first person person A, the second person person B, and then C, and then D, and then E, and it's also going to be 120. Or how many ways can five different books be arranged on a shelf? Now, there is a bit of a trigger word here. What I'd like you to do is underline or highlight the word arranged. The word arranged is what tells you it's a permutation, specifically on a test. If you're going, I'm not sure, is this a permutation or a combination? We haven't done combinations yet, but you will by the time you get to the test. If I use the word arranged, arrangement, it's a permutation. So how many ways can five different books be arranged on a shelf? 120. How many permutations are there of the letters of the word prove? Look at the word prove. Is each letter different? Then that's the same as A, B, C, D, E. Now, next class, we're going to look at what if you have letters that are repeating? What if you've got like Mississippi with lots of I's and S's and P's, and we'll deal with that, and it's actually a fairly easy twist tweak to do. But today it would be 120. Let's go back to our five letters, A, B, C, D, E. Instead of using all of the letters, we could use fewer letters. For example, D, B is a two-letter permutation of these five letters, as would D, E, B, and C, D, and C, B. How many of those are there? How many letters are they asking us to pick? How many choices do I have for the first letter? How many choices do I have for the second letter? How many two-letter permutations are there? Twenty. We call this an arrangement of five letters taken two at a time. How many different three-letter permutations are there? Five times four times three. Twenty times three. Sixty. So, this was typed up in 2001 before iPods and MP3 players. This was back when CDs were still the rage. But you can just think of this instead of a CD. Think of your iPods scramble feature. So, when you press the scramble button on a CD player, it plays a permutation of the song. It scrambles the order, but it plays each song once before moving on. So, it's not going to repeat songs. If your CD has five songs on it, how many permutations of the song are possible? Five times four times three times two times one. Hundred and twenty. How many permutations are there of the word compute? How many letters are there? Seven. Are they all different? Then it would be seven times six times five times four times three times two times one. And we're going to do this for the last time by hand. And then we're going to say, oh, for Pete's sake, someone has come up with a quicker way to go. A number times one less times one less times one less times one less times one less. Down to one. What is seven times six times five times four times three times two times one? Sorry? Five hundred and forty? Five thousand forty. That seemed way too small to me. Five thousand forty. We're going to introduce something called factorial notation to you. And the symbol for factorial is an exclamation mark. You don't read this as one. You read this as one factorial in there. This is not two. This is two factorial. Three factorial. Four factorial. Five factorial. And factorial notation. Well, five factorial means five times four times three times two times one. Four factorial means four times three times two times one. Three factorial means three times two times one. One factorial is defined as one because you can't go one minus and one. And it's built into your calculator. So we're going to look at this five thousand forty. All of you, all of you, all of you get your calculators out. Turn it on. And you're ready. Now we get to go into the wonderful math button right there. Press the math button. And we're doing probability mathematics, which is you can either go right arrow three times, or if you go left arrow once, you end up in the probability menu, which is faster. And what's option number four? Now, we were silly. Hit clear, clear, clear to do seven factorial. First you have to hit seven. And then go math, left arrow, option number four. And if you hit enter, five thousand forty. What's ten factorial? You know what? There's thirty of you in this classroom. How many different ways, how many different seating arrangements could I make? What's thirty factorial? Yeah, okay. And this is for, I'm doing this on purpose to follow along. What's forty factorial? What's fifty factorial? There's going to come a point when your calculator crashes. And that's going to tell us how big a size we can have before our calculator can't handle it. Sorry? Oh, someone's played this already. Okay, yeah, seventy factorial crashes your calculator. We can't have seventy objects. Well, that's a lie. We can, but we're going to have to be sneaky about it. Because there are plenty of situations where you do want to mix up more than seventy objects in the real world. So surely there must be a way around it. Yeah, seventy factorial will crash your calculator. Okay. We can have sixty nine objects. We just can't have seventy. That's the limiting factor. Okay. But this is factorial notation. Very nice shortcut. Especially, you noticed yesterday on our last lesson in the fundamental counting principle, how often you were going one number times, one less times, one less times, one less times, one less times. Now the only problem with factorial notation is it always goes all the way down to one. But here in example three, we still want it to go down by one, down by one, down by one. But we didn't want to go all the way down to one. I wonder if they got that built in and they do. Okay. What I want to talk about is the algebraic version n factorial. If you expand this and you're going to need to be able to, this is really n times one less times one less than that times one less than that. And we're going to go dot dot dot times two times one. That's the algebraic expansion of n factorial. n times n minus one times n minus two times n minus three times n minus four times n minus one. Quit when you get to one. And you're going to be asked to do some algebraic expansions of these. So we're going to try a couple of more here. I'm going to put a little b right here. What would n plus three factorial be? Well, it would be n plus three because that's the first number. What's one less than n plus three? n plus two, n plus one, n dot dot dot dot. In fact, you know what? Instead of writing n dot dot dot dot, I could just, if I was really clever, I could say, because this is an n factorial, n times n minus one times n minus three times n minus three times n minus four. And sometimes you'll want to do this. Sometimes you'll want to go further. Sometimes you won't want to go as far. It's going to depend on the question and I'll give you some examples. Hey, what would n minus four factorial look like? Well, it would be this first number. What's one less than n minus four? It's not n minus three. Careful, what's one less than n take away four? n take away five. By the way, what's the most common mistake? n minus three. What will be on your multiple choice? n minus three, absolutely. n minus five, what would the next one be? n minus six then? You know what? I'm going to take a shortcut. I'm just going to write n minus six factorial because that is n minus seven, n minus eight, n minus nine. Why do you stop there? I'm stopping wherever I want to right now. You'll see there is a good place to stop eventually. Example three. So, factorial notation, very nice. Certainly a good time saver. Would you really want to go 30 times 29 times 28 times 27? And do you think you wouldn't miss one to get that? What was it? It was times 10 to the 32 when all was said and done, right? How many three-letter permutations can be formed using the letters of the word compute? Well, how many letters are there in my grab bag for the first one? Seven. Then? Then? I'm going to argue that this is really seven times six. Actually, you know what? I'm going to give you the short version first. I'm going to argue that this is seven factorial divided by four factorial. Think about it. And then I'll expand the top. Seven times six times five times four times three times two times one all over four times three times two times one. Can I cancel? This is factorial, a factor. In fact, really I could have just done this, Miguel. Seven times six times five times four factorial over four factorial. Do you remember I said to you, you'll know when to stop writing depending on the question? Here, I would stop writing there because I would say, I wanted to cancel. Which I would argue, I mean, seven times six times five, that's shorter to type by hand. But if it was 30 letters, it would probably be shorter to go with some kind of a factorial notation on a calculator. Or it would even be better if they somehow had this built in. And they do. By the way, how many letters grand total? Seven. How many did we choose? Three. You know what? I could think of this as seven factorial all over seven minus three factorial. The total number of letters divided by the total number of letters minus how many you're choosing. Or how many are left? That's where that four came from. We're trying to set up a formula here, which is why I'm going with smaller numbers and you'd be saying, I wouldn't do it that way. I know. But for bigger stuff, we're going to try and set up an easy formula so that we can fall back on it. In fact, the notation is this. Seven p three. This means permutate seven objects three at a time. Mathematically, it means seven factorial all over seven minus three factorial. Yes, four. And it is also built into your calculator. Turn your calculators on if they've turned off already. We're going to type this. What we're going to type is seven math button back to the probability section. See the NPR, that's the N, the total number, permutate our objects at a time. So we're going to pick option number two and put a three here. And if you hit enter, that'll give you the same answer as we got before. Oh, did we fill out the 210 last time? I don't think we did, did we? Answer is 210. Whoops. 210. So this class has 30 people. How many different arrangements of four people at a time could I make? It would be 30 math back. Option two, 30 p four. Or you know what better yet? How many people are at each table in my class? How many different lab partner combinations could I possibly come up with? Stop. You know what? Why don't I just go second function enter? It's way faster. And you want to find some shortcuts for typing because there is a lot of typing. And you'll get quick at it, but it is a lot of typing. Apparently 870. Although that counts Justine there and Pat there as different from Pat there and Justine there. I guess technically I would divide this by two if I wanted to find the total number of lab partners. Because Kyle, you don't care whether you're listed first and Troy is listed second or vice versa. Or maybe you do, but we wouldn't count it that way separately. And again, we're ready. I mean, certainly with this answer, we're well into this. No way we can count that by hand. Here's the generic one from n objects permutate r of them. It's defined as n factorial all over n minus r factorial. And that's on your formula sheet. Check it out. See it there, Stephen, right? But I think two thirds of the way down the page. Yes. So let's summarize. An ordered arrangement of distinct objects is called a permutation. What do we mean by ordered? We mean the order makes a difference. Kyle sitting left, sorry, Kyle sitting right of Troy is counted as different as Troy sitting right of Kyle, even if they're at the same table. If you have n distinct objects, if you're taking all of them, there's that many. That was where we went all the way down to one. If you're only taking a subset of them, if you're not taking all of them, oh, the number of permutations of n distinct objects taken r at a time, we write it as n p r and it's that n factorial all over n minus r factorial. And it's built into your calculator because it's built into your calculator. They will ask you just to calculate it. Like you'll see how many different ways can you arrange three books from a group of 15 and you'll go 15 p three on your calculator. That's the classic multiple choice and they'll give you that. But because that's so plug and chuggish, also what they'll ask you to do is some factorial algebra simplifying like this says write each expression without using the factorial symbol. So if they give you n p two, I would say that's n factorial all over n minus r r is two n minus two factorial. And what they want me to do is they want me to start to expand this expand this similarly to what we did right here. Remember, I said you'd know how to stop. This is where you'll know how to stop. First thing I would do is I would look at the top and I would look at the bottom and I would ask myself which one is bigger n or n minus two. Ian, that's the one I'm going to expand. I'm going to write this fraction line. I'm going to write the top one as n times n minus one times n minus two. And this is how I know how to stop. Do you see that the term on the bottom just appeared on the top? I could write times n minus three times n minus four times n minus five times n minus six times n minus. But really, that is just that. And the n minus two factorial is still in the denominator. Can I cancel? Is it factored? Oh, I should do it because that's why you did that big song in dance class. Do you know what np two simplifies to? n times n minus one. Let's write that as a separate answer. In other words, remember a few minutes ago we did, hey, 30 objects, 30 students taken two at a time for our lab tables. And we went 30p two, 30p two. I could also apparently have gone 30 times n minus one and gotten the same answer. Oh, it works. Or they'll give you, by the way, they may write this as n squared minus n. They may multiply the brackets out. I won't because I think that looks cleaner to me. Or they may give you just not a permutation. They may just give you some kind of factorial algebraic expression. That's fine. I start out the same way. Victoria asked myself, which one's bigger, n plus two or n minus one? That's the one I'm going to start to expand. n plus two is actually n plus two times what's one less than n plus two? n plus one. Oh, keep going. I haven't got this yet. What's one less than n plus one? n. Keep going. What's one less than n? Ooh, ooh, n minus one. And this is how I know to stop because isn't that what's on the bottom? I could also keep going, though. And I could go n minus two, n minus three, n minus one. But I think I'm going to argue that's the same as just n minus one factorial. And I have an n minus one factorial on the bottom. Jesse, can I cancel? You're supposed to say, yes, I can. I guess this one simplifies to n, n plus one, n plus two. That's probably the order we would write it. We usually go from math nine, smallest term to biggest term. But if you wrote any order, I'd give you four months. They might ask you to foil that out. I wouldn't. I think they do in the book, and you can to see if you're right. But to me, this is a much cleaner answer than foiling that out and then multiplying and n and then getting n cubes and squares and blah. Because this I could do in my head. I could say, well, tell me what n is. 10? Oh, this is 10 times 11 times 12. I can do that. Where if it's foiled out, it's got cubes and things I can't do. How many terms on top? Two? How many terms on the bottom? You know what, Justine? I'm going to mentally just draw a line down the middle that covered up too much stuff. But I'm going to mentally draw a line down the middle. I'm going to treat this as two little mini fractions. I'm going to simplify n over n plus one. Sorry, n factorial over n minus one factorial. And I'm going to simplify n plus one factorial over n plus two factorial. So I'm going to draw a great big fraction line. Looking at the first two terms, which one's bigger? I'll start to expand that. n factorial is n times n minus one. Oops, I'm going to stop there. And I'm going to rewrite the n minus one factorial from the bottom. And I'll come back to this. Now I'm going to do the second half of this fraction. Matt, which one's bigger? n plus one or n plus two? Oh, I'll start expanding the bottom one. n plus two would be factorial. Would be n plus two times n plus one. Ooh, I'll stop there because that's what I wanted to appear. So it would cancel out on the top. Now I can go cancel happy. Those cancel. Those cancel. You know what this whole monstrosity simplifies to? What's left on top? What's left on the bottom? So on the provincial, now this test will not have a non-calculator component. But they could make this a non-calculator question on the provincial. They could say give you this and then they could say evaluate for n equals five. In that form, I don't think you could do it for n equals five. It would be very, very yucky. But if n equals five, I guess the answer is five over seven, which I can do in my head showing no work. Or they might just want you to leave it like that. I will put one of these on your multiple choice section of your unit test. They can give you a factorial to simplify. Or they can give you a permutation equation to solve. Now this says xp2 equals 42. Remember that npr is n factorial all over n minus r factorial. Looking at this, I'm going to fill this in for this. I'm going to say this is apparently x factorial all over x minus 2 factorial. That equals 42. We can't cancel out exclamation marks. We can only cancel out identical factorial terms. What we're going to have to do, Dylan, is we're going to have to simplify the left-hand side to get rid of the factorials. And renew that by asking ourselves, what's bigger, x or x minus 2? x, we're going to start to expand it. This is really x times x minus 1 times x minus 2. Justine, here's where I'm going to stop. I could keep going, but I'll put the factorial right there all over x minus 2. Minus 2 factorial, that equals 42. Can I cancel? Yes. And in fact, I end up with x times x minus 1 equals 42. I end up with x squared minus x equals 42 when I get rid of brackets. Amy, what kind of an equation is this? It's a quadratic. How do I know? It's got a squared. How do I solve a quadratic first thing before I do anything else? You've got to make it equal to 0. Factor it. They will always factor nicely. And in fact, you'll get the nice trinomials that don't have the thing in front of the squared. You're looking for two numbers that multiply to negative 42 and add to negative 1. x minus 7, x plus 6 equals 0. What are my roots? Problem. Can I have 7 objects? Yes. Can I have negative 6 objects? Try going negative 6 p2 on your calculator. And yes, once again, we reject. Whose date was I making fun of in this class? Oh, Matt. Sorry. Did you see him? Me! Me! Make fun of me! No problem. You're going to always get an extraneous root on these. I'm going to put a guarantee you on the written section. I'm going to ask you to solve a factorial equation. Partly because what a lovely quadratic you get. And you will get a quadratic most of the time. Partly because the nice little review combination of stuff. B is a bit different. We're going to come back to it. Let's try c. So I would say x p2. x p2, that's actually when I expand permutation. That's x factorial divided by x minus 2 factorial. That's what the letter p stands for in that formula. Equals 110. Which one's bigger? The top one. This is going to be x times x minus 1 times x minus 2. Stop there, Mr. Dewick, because that's the term you want to cancel on the bottom. Equals 110. Can I cancel? It is factored. You'll get x, x minus 1 equals 110. You'll get x squared minus x equals 110. You'll get x squared minus x minus 110 equals 0. Numbers that multiply to 110 and add to negative 1. Oh, 10 and 11. x minus 11, x plus 10 equals 0. x equals 11, negative 10. Sorry, Matt. So you can't do a negative permutation, which makes sense, because what the heck is negative 6 objects? B. I'm covering B because I've never seen it on the provincial, but I don't know why. I am not going to put it on a test. I've read the exam specs, and it depends on how you interpret the exam specification, but I don't, this could show up. So I'm going to show it to you. This is where the x is on the right-hand side, but I won't ask you this on a test. I'm still going to expand this, Jesse, by saying, well, 10 px, that's n factorial over n minus r factorial. That's going to be 10 factorial all over 10 minus x factorial. That equals 90. And there's a problem because there is no mathematical skill available that can somehow multiply the exclamation point into there. So you cannot pull stuff out of a factorial. You can only expand it by going one less, one less, one less. And what's one less than 10 minus x? Well, I guess sort of 10 minus x minus 1, but that doesn't help me much because that gets even yellier. Instead, here's what we're going to do. We're going to, first of all, cross multiply to get the x on top and the 90 on the bottom. I'm going to move this up to here. I'm going to move this down to there. My physics 12 students are going, oh yeah, we've been doing this like crazy. What I'm going to say is this is the same as 10 factorial equals 90 bracket 10 minus x factorial. But Amy, I don't want to leave the 90 there. I also want to divide by 90. I want to move the 90 over to this side. What I really wanted to do was to get all of my numbers to one side and all of my x's to one side. Why? Ah, watch. What is 10 factorial if I start to expand it? 10 times, okay, I'm going to start to expand it. Don't write this down yet. Just watch. If I start to expand this, I'll get 10, oh, this is over 90 and this is 10 minus x factorial. I'll get 10 times 9. Ooh, why did I stop there? Can you see I have a 90 on the top and on the bottom in disguise? See it? Can I cancel? Is it factored? In fact, I get this. 8 factorial equals, now, if this is 8 factorial, this also has to work out to 8 factorial. What number would I put in for x so that I could go 10 minus something and get an 8 factorial equals an 8 factorial? x must be 2. That's so weird because I solved it all indirectly. Did I actually solve for x? No, I solved for the other side and figure out what x had to be to make the other side work. So I've showed you this where the x is right there. Now, stop. How many lines? How many lines of work? Please count. How many lines of work? Seven. If it was multiple choice, if this was a multiple choice question, don't do seven lines of work. Get your calculator out and go. The answers they would probably give you would be 6, 7, 8, 9. Go 6p2 equals, not 42. 7p2 equals, oh, 42, done. If it's a multiple choice question, plug the answers into your calculator, please. And if they gave you this as a multiple choice question, they would give you answers probably of 6, 7, 8, 9. I'm going to guess. Plug the answers in. I will give you one of these guys on the written, but I'll probably give you one of the multiple choice, too. If that's the case, please don't waste your time. Just plug in the answers. What is zero factorial? Try it on your calculator. It's not zero, which is kind of, huh? Not zero. What is it? It's defined as one. It's programmed in. Why? How come? Well, if I go from five objects, permutate all five of them, we can do that two ways. If you're taking all five objects, it's just going to be five times four times three times two times one. It's supposed to be five factorial. But if we use the permutation formula, this would be five factorial all over n minus r factorial, which would be five factorial all over zero factorial. For five factorial over zero factorial, to give you the same answer as five factorial, what had that bottom better be? What's on the bottom of this fraction over here, even though it's invisible? One. It's got to be, we have to define it as one. We have to. Otherwise, our whole system break down. And I know it doesn't make sense, because you're going, wouldn't it be zero minus one, or wouldn't it be a zero somewhere in there, and anytime zero is, this is the one time where anything times zero is not a zero. We defined it as one. Otherwise, our original method no longer works. In case you wondered, we're going to try a couple more from the workbook, and then I'll turn the loose. By the way, remember I said last day, you can always fall back on the fundamental counting principle. You can drawing blanks. I'm good at drawing blanks. You can drawing blanks. But you can also practice the shortcut, the permutation shortcut, and the factorial shortcut. So, example one, lesson two, page, what did I say it was? 383. It says, find the, we're going to do example one right here. Find the value of 43 factorial over 40 factorial. Look, this is really going to be 43 times 42 times 41, and then there would be times a 40 times a 39, times a 40 factorial, but that would all cancel out on the bottom. And this is also how, Ian, we can do objects bigger than 70 because we can cancel out a bunch of the terms, and we'll get an answer that won't crash our calculator. So, the permutation, the P menu, that your calculator can go bigger than 70. In fact, try it. Try going 80, P5. It won't crash your calculator. I don't think. I hope not. 80, math, P5. Yep. I can get more than 80. I can now do more than 70 objects. If you do it the long way using the formula and the factorials, anything more than 60, anything more than 69 will crash your calculator. But with this, it does the dividing first. They thought of that. We'll come back to that one. Simplifying, we've done a few of these simplifying ones. I'm going to give you some of those for homework. I want to just try a few applications. So, if you would like to turn the page to page, example three, how many permutations are there of the letters in the word Regina, the capital of Saskatchewan? Are all the letters different? Are we picking all the letters? Yep. How many letters are there? This is seven factorial, which is what? Oh, it's also seven P7, but I think it's faster to go seven factorial. Less typing. That's what I said. Six factorial, which is what? Whoops. What's six factorial? Six times five times four times three times three times one? Seven hundred and twenty? Kolona. That's the one I was thinking of. Seven factorial. By the way, it's faster to go second function, enter backspace and change the six to a seven than to go seven, math, arrow, down. The only thing I don't like about the TI is they put these in a very, very obscure sub-sub menu. It's a lot of typing to go find them. Sorry. Oh, what is it? 5040. Here's our definition of what a permutation is. So let's go to example five. In a South American country, vehicle license plates consist of any two different letters followed by four different digits. Find out how many different license plates are possible by the fundamental counting principle and permutations. So using the fundamental counting principle, I'm going to go two letters. One, two, three, four digits. How many choices do I have for the first letter? Twenty-six, and it says they have to be different. So how many choices do I have for the second letter? Twenty-five. How many choices do I have for the first digit? Ten. And they have to be different. So then I have nine. I have eight. I have seven. Six times 25 times 10 times nine times eight times seven. And this is my fallback. I can always hopscotch my way there, blank by blank by blank by blank. What do you get? Read me the digits three, three, two, seven, six, zero, zero, zero. Now I could also get there by permutations by two separate permutations. How many letters are there in the alphabet? Twenty-six. How many am I permutating? Two of them. Times. How many digits are there? Ten. How many am I permutating? Four at a time. I think if you do that, try this. You'll also get 3,276,000. Troy, is that wrong? Oh, okay. Yes? Which method's better? They're both fine. Which method's easier? This I see way better. I'm still trying to get good at the shortcuts. I would probably try the shortcut, get an answer, and then somewhere on a scrap piece of paper or in the margin, I draw my blanks, use my Scrabble method to get the same answer and go, yeah, I'm right. What's your homework? I don't know. Let's see here. Three A, B, and C. Five A, five B. Now five B, do you notice how you have factorials on both sides? I don't like that. I always want all of our factorials on the same side so that we can then ask which one's bigger and cancel. The first thing I would do is I would divide by N minus 1 and move it over here. Okay. And five C. Six is good. Nine is good. 12 is good. We'll pause there. Permutations.