 So Block D, let's talk a little bit about how things are going to end up here. So I'm aiming towards your test being Wednesday next week. When I walked into my Block B class this morning, I had about four students in the front saying they found this stuff pretty easy. Most of it is, I think today is where we're going to be throwing at you not exactly a curveball but a fair bit of calculator work and a question that it's really really easy if you try taking shortcuts to make sloppy mistakes. You're going to notice later on in today's lesson I go very, very, very step-by-step. In fact, you're even going to see me write ones in places where I normally say to you I'll never write a one there. This is about the only time where I'm going to put a one in certain location. So ideally we're heading for test Wednesday. Today I have another take-home quiz for you which will be marking this Wednesday. Then I have a great big unit review take-home quiz for you that should be going over on Monday. Okay? So answer the following, showing your work. One mark each unless otherwise indicated. Mr. Deweyck, it says, is purchasing a new Lamborghini sports car. I wish. He has a choice of, oh, okay, this is going to be six body colors, three body styles, four interior packages and three tire styles and oh, 216 I think, yes? 216. On your test, especially the multiple choice portion, it'll roughly mirror the order that I've taught you things. In other words, remember the first thing we learned was the fundamental counting principle. That'll be the first couple of questions on your test. And then we learn permutations. That'll be the next couple of questions on your test. And then we learn permutations with letters repeating. That'll be the next couple of questions. Oh, and then we did combinations. That'll be the next couple of questions. And then probably a pathway question. Oh, and permutations, we also did factorial simplification. It's a factorial stuff. That'll be there as well. That'll go roughly in the same order. That might help you in your strategy. If you're on number 18 multiple choice, probably not a permutation and likely could. How many ways are there to arrange the letters of the word complainers? Are any letters the same? Nope. One, two, three, four, five, six, seven. Is it 11 factorial? Which is, I have no idea. 39916820s. 39916820s. 39,916,800 different ways to arrange the letters of the word complainers. Now, the other way you could have got this is you could have gone 11p11, but I think it's faster to go 11 factorial. But it is from 11 letters, permutate all 11 letters. How many seven letter arrangements are there of the word ambidextrous? One, two, three, four, five, six, seven, eight, nine, ten, eleven. So it has 12 letters. But we only want to permutate seven at a time. So you could either go 12p7, or you could go one, two, three, four, five, six, seven, use 12, then use 11, then use 10, then use nine, then use eight, then use seven, then use six. Or 12p7 actually is 12 factorial all over five factorial. It's 12 factorial over 12 minus 7 factorial. The fastest way in terms of typing is this one. But all three of these, if this was multiple choice, would be legitimate answers to pick from as well. But in this case, they wanted us to actually evaluate 12p7. Oh, same number, but just one zero less odd coincidence. Three million nine hundred and one thousand six hundred and eighty. How many three letter arrangements can I make from the word campgrounds? Okay. All the letters are different. One, two, three, four, five, six, seven, eight, nine, ten, eleven letters. So I can either go 11p3, or I can go 11 times ten times nine. Or I think looking at my answers here, they want me to find the NPR equation on my formula sheet and substitute in the numbers. I think it's going to be 11 factorial divided by eight factorial C. This would be an example step of how they could give you a combinator's question on the non-calculator section of the exam. Because of course, they can't ask you to get these numbers in your head, but they can certainly ask you, I think 11 take away eight or 11 take away three is fair game to do in your head. How many four digit numbers can I make from the following group? Well, my students this morning, they made a valid point. They said, Mr. Dewick, this question is unclear because you're not sure whether you're allowed to repeat or not. I said, you're right. It's not. It doesn't say whether we're allowed to use the same number or not, and it should. It will on the test. So I'm going to take both answers. I'm going to say without repetition, I would go one, two, three, four. How many choices do I have for the first number? Why not six? Okay, so I'll put a five there and let's say I used up an eight. They've already drawn the grab bag for me, which is kind of nice. How many choices do I have now? Five, four and three. In fact, it's 20 times 15. So I will accept 300. Or if you answered with repetition, one, two, three, four, you would say I still have five choices for the first number, but then it's six. Six and six because I keep throwing the numbers back into the bag and starting over. You're right. This question is not clear. I actually said to my last class, I'll change it. Then I thought actually last block, maybe I shouldn't change it because maybe the discussion of whether it's repeating or not is actually worth having it be unclear so that we get used to having to look just before the test. So you know what? I think I'm going to leave this for next year and have the kids again raise the same point. Hey, it's not clear. It's a good discussion. One of the few times I'll leave a mistake on their own purpose. 1080, is that right? What a good HTV is. That's correct. Okay. Say that again. Say it one more time. I hope you all picked up on a hint. I have one question in mind and it drives me crazy and I phrased it trickily on purpose to try and tempt people to start with. No. We haven't started with a zero the whole time. Oh, it drives me crazy. Mississippi, now we got repetitions. One, two, three, four, five, six, seven, eight, nine, ten, eleven. It's going to be eleven factorial all over four i's, four s's, two p's. Oh, you know what? I'm going to go second function, enter, second function, enter. Second function, enter. There's my eleven factorial. I already got it there. Divided by, you know what? I don't want to go four factorial, four factorial two. That's a lot of typing. I know four factorial is twenty-four. So this is twenty-four times twenty-four times two. I think that's faster to type. But there are, there is more than one number in the bottom, Madeline. So I better put it in brackets, right? Thirty-four thousand six fifty, is that right? People nodding and yawning? Yes, you. I didn't apologize, just pointing it out. Uh, Chillowack. One, two, three, four, five, six, seven, eight, nine. Ten letters? Yeah, that's right. Ten factorial all over. Two c's, two i's, two l's. By the way, I'm not going to type two factorial, two factorial, two factorial. You know what I'm going to type in the bottom? Yeah, eight. Second function enter. I'm also going to cheat some that have an 11 factorial sitting right there. Ten. Eight. That probably didn't need the brackets, but whatever. Four hundred fifty-three thousand six hundred? And we're eight. Pathways. Is this a regular or an irregular pathway? Yeah. So you can use the factorial shortcut, which is what I'm gonna do. You can use Pascal's and they'll get you there. Pascal's works for anything and it's nearly cool. I'm going to go right, right, right, right, down, down, down, down, down, down, down. I'm getting 10 factorial all over, 10, 1, 2, 3, 4, 5, 6, 7, how about 11 factorial, Mr. Dewick? All over 4 factorial, 7 factorial. What I said last class, Chelsea, there is a built-in shortcut here. Do you notice that the 4 and the 7 add to the 11? That just happens as a wonderful coincidence to be 11 choose 4, which is less typing. Now that shortcut step only works if those bottom two add to that, but if you look at the choose equation, the choose equation is this. So I'm going to type 11 choose 4, it's less typing. 11 choose 4, 330, is that right? Just curious, anybody do Pascals? Triangle? You got the same answer? And you know, Greg said I did both, you know, probably on a test I would double check my answer and do both, and if I got the extra time, why not, okay? So this quiz is worth, count them eight marks, give yourself a score out of eight. Make sure your name is on it. Page is 416, I think, isn't it? Page 416, Pathway Problems, oh, 419, sorry, 419, Pathway Problems. I'm going to do K, but I'll say are there any others you would like me to do, yeah. D is in dog, J as well, yep. So D is, did you just figure it out? Because you got to come up or back, right? So you have one more thing on top. And I said to you, I'm not going to do, I'm not going to do a 3D1 on your test. I've never seen it on the provincial, but I don't know why, because they're nerdily cool and when I, there's a specific exam learning outcome and the way they phrase it, Chelsea, it doesn't specifically say two-dimensional pathway problems, it just says using pathway problems. So I've covered it just in case, yeah, I guess. So J and K, who asked J? Okay, I'm actually, because I can do at least this part, I'm physically going to flip my screen so that the A is sort of above the B. I'm going to do this and work this way. Okay? Because you guys would tilt it 45 degrees anyways. I'm not joking, I really have a tough time doing these upwards, I think downwards with Pascal's. So I've done that. And I'm going to zoom in when I can find where my zoom thingy is, this is going to be a little weird. Oh, you know what you did? You did it the other way? Oh, you went this way? You went this way? Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Wrong K. Example K. What bugged you about this? This. I really blame, was not, I, what, oh wait a minute, Mr. Dewick says if I want to, I can start on the other end and work my way this way and I'll get the same answer. I'm going to start at B and work my way to A. I think it's going to be way easier. That's my, that was my first decision because I really wasn't sure what to put here really. So Chels, I started here and I went one, one, one, one, what number goes here? Now by the way, let's talk about what we're really saying. What we're really saying is there's one choice to get there. One choice to get to that corner, but you have two choices to get to this corner. We're adding up the numbers, but we're really adding up as a number of different pathways to get there. That's why Pascal's works. It's going to be a one here, a three here, blank, oh wait a minute, I missed a row. I should have filled that one in there as well, my bad. What's going to go here? A one because it's one plus zero, one, a one there. One, four, four, two, one, five, eight, six, three, one, thirteen, fourteen, nine, four, and this four is a dead end. How many ways can you get to this dead end here? Four different ways, but then you're stuck there. I think by the way, if you went from A to B in this pathway, you would have found in J zero there. You would not have been able to get there at all. You never would have put a number there. Let's keep going, twenty-seven, twenty-three, fifty, oh, what's going to drop down to the A? Twenty-three plus, what number's right here? It's invisible. I'm going to quit, but apparently A is twenty-three, is twenty-three the answer? No, I would have done that. If you did it the other direction, you'd go one, one, one, and you'd keep going from there. You'd get a two there, and you'd be dropping down a two, and you'd end up leaving this whole section blank because you can't get to it at all. But I figured why not start with the prettier part because I can figure out how to start, and then I just have confidence with the dominoes and the lines. Pathway problems. Nerdly cool. Today's lesson has one of the most impressive sounding names in all of Math 12. They are going to teach you the binomial expansion theorem. Leslie, honey, what did you learn in school today? Mummy, I learned the binomial expansion theorem. All of you today at supper time, when your folks say, what did you learn in school? I learned the binomial expansion theorem. And that's a fancy name for a shortcut for foil with a vengeance. This is the last lesson, but it'll probably take me two days. Lesson six. Lesson six. And this is a great shortcut. What I want you to be doing for the first few minutes is see if you can spot a pattern. And Martin, you're going to sit up for me on my friend, and here's what this says. First of all, we need to remember a binomial, hey, that's a binomial. And what we're going to ask ourselves is, is there a shortcut for foiling? We're going to do a plus b squared by hand. We'll do a plus b cubed by hand. And then we're going to start to see if we can spot a pattern or a shortcut. Because on your test, I may say to you, do a plus b to the 19th power, which is by hand next to impossible. Because here's what by hand means, a plus b squared, write this down. That means a plus b times a plus b, right? And then you would do good old foil, and when you do foil, when you multiply this out, you'll get a squared plus 2ab plus b squared. That's math nine. If I want to do a plus b all cubed, that would be a plus b times a plus b times a plus b. And to evaluate this, Chelsea, I would multiply the last two, get an answer, and then multiply the first one in. But I'm just going to tell you what the answer is, because we're looking for a pattern, and also to prove to you that it is possible to sort of cheat and do these in your head. It's going to be a cubed plus 3a squared b plus 3ab squared plus b cubed. It is. I'm going to say, hold your questions. You'll spot a pattern probably with about two more examples. A plus b to the fourth, now write this down. That would mean a plus b times a plus b times a plus b times a plus b, and there is no shortcut to do this all at once. You would have to multiply the last two, get an answer, multiply the third one in, get an answer, multiply the fourth one in, get an answer, gather like terms. You see, Kellan, one of the things with multiplication, if I want you to add 10 numbers in a row, it's easy. You write all 10 vertically, and you can do the entire column at once. The entire column at once, there is no math trick for multiplying four things in a row. You can only multiply two at a time, which would become very tedious. Oh, I'm going to tell you the answer, by the way. This is going to be a to the fourth plus 4a cubed b plus 6a squared b squared plus 4ab cubed plus b to the fourth. How am I doing it that fast? So we're looking for patterns now. You ready? Oh, write that out, first of all. And the first question I ask you is, what patterns do you see in the powers of A? What's my overall initial exponent in the first example here? Two, what do you see in the powers of A? Two then, two then, one then, trick question, zero. What's the exponent right here? Three. What do you see in the powers of A? Three then, then, then, trick question, zero. So what pattern do you see in the powers of A? Decreasing by one. What pattern do you see in the powers of B? I think exactly the opposite. What power of B is in the first term here, trick question, zero, then, then, two. What power of B is in the first term here, zero, then, one, then, two, then, then. So start from zero, increase by one. Now Blaine, that doesn't help me get the coefficients, doesn't help me get the four or the six, but for A plus B to the fifth, I do know this. Blaine, what's my exponent? It's going to be A to the fifth, A to the fourth, A cubed, A squared, A nothing. It's going to be nothing, B to the one, B squared, B cubed, B to the fourth, B to the fifth. I'll come back down in a second if you didn't quite get that, although maybe you can fill it out on your own just spotting the pattern. Can anybody see the pattern of the coefficients? I'll put some ones where I normally don't put ones. Justin, what? Are you saying this is Pascal's triangle? Was there a row one, two, one, and was the next row one, three, three, one, and was the next row one, four, six, four, one, and ooh, ooh, ooh, okay, you got it here? One plus, what's going to go here? Five plus, what's going to go here? Ten plus, ten plus, five plus. I'll put the one in there to spot the pattern. Oh, by the way, Martin, I'm going to suggest to you that that is way, way, way, way easier than writing A plus B out five times. Multiplying the first two, getting an answer, multiplying the third one, and getting an answer, multiplying the fourth one, and getting an answer, multiplying the fifth one, and getting an answer, and gathering like terms. Oh, but there's an even better shortcut, because what if, Hannah, I said A plus B to the 12th, and you didn't have the previous 11 above? Do you remember the terms in Pascal's triangle were also chooses? Remember? Hannah, what's my exponent? Five. Actually, even better than Justin's noticing that this is Pascal's triangle is to realize this is five choose zero, and you know what this is? Five choose one, and you know what this is? Five choose, except I need to make the C big, five choose two, five choose three, five choose four, and five choose five. That's actually kind of a nice pattern, too, because Madeline, what's my exponent? What's my choose? Five. Oh, remembering that I have to start counting from zero, zero, zero. Oh, what's my exponent? Five. How many terms are there? Count. Six. Why are there six terms in the exponents of five? Because I have to start counting from zero. So here's a question I've actually seen on the provincial exam. A plus B to the 18th has how many terms? 19. One extra. So do you see the pattern for the numerical coefficients? It's going to be n choose term number. Oh, wait a minute, that's not quite right, because what term number is this? This is term number one, but what choose is this? Choose zero, and what term number is this? Term number, but what choose is this? Choose, you know what? It's n choose term number minus one, I guess, because we got to start counting from zero. So we can take this using the A and the B, and we can really expand it. Don't write this down. Just watch. If they said find the, let's go 4A plus 3B to the 12th, and they said write out the first four terms. You know what? Instead of a B, let's put an X and a Y there, because that's what you guys are used to. I would, if I was doing this question, and this will be a test question, by the way, probably I'll ask you for four terms, and I bet you'll make it worth two marks, which means what's each term going to be worth? Half mark, something like that, or I'll make it worth four marks, probably two. I would say, hey, that's A, that's B. What's my exponent? Okay, so it's going to be A to the 12th, A to the 11th, B, A to the 10th, B squared, A to the 9th, B cubed, there's the first four As. By the way, can you see the built-in error check? What do the exponents always add to? 12, is it? Okay. What did you say the exponent was? So 12 choose 0, 12 choose 1, 12 choose 2, 12 choose 3, but then I would also fill in what A and B are. 12 choose 0 is 1, there's a reason why I said that was worth memorizing it. It was really for this question. Just sitting where the A is, it would be 4x to the 12th, which means when I tidy this up, I'd have a 4 to the 12th and an x to the 12th, plus, you know what 12 choose 1 is? 12, that's why I also gave you that particular shortcut. And it would be 4x to the 11th, 3y to the 1, plus, I don't know what 12 choose 2 is. Can someone find that for me, please? Anyone, anyone? 12 choose 2. This is going to be 66, 4x to the 10th, 3y squared. And what's 12 choose 3, 220, 4x to the 9th, 3y cubed. Now I wouldn't quite be done because this one, the entire coefficient is going to be with this first term, 1 times 4 to the 12th. I might get a scientific notation number here because I made this one up. I don't know. Oh no, it works. It's big, but it works. It would be 16777216, 16777216x to the 12th, to the 12th, Mr. DeWitt, plus, what would the coefficient here be? It would be a 12 times, can you see the 4 to the 11th? See it, see it, see it, see it. Times, see the 3 to the 1, 150994944, 150994944. And then my variable would be x to the 11th, y to the 1. And yeah, you sometimes get big numbers. Plus, what would this one be? This would be 66 times 4 to the, see the 4 to the 10th hidden in the question? Times 3 to the what? 3 to the 2. The coefficient is going to be that. And then I'll have an x to the 10th, y squared. I'm not going to round it out because we're going to do a couple like this. Oh, and then the last one would be a 220 times, there'd be a 4 to the 9th times and a 3 to the 3rd. There's your coefficient, x to the 9th, y cubed. Way faster than foiling that out. Let's start out doing some smaller ones. So it says, visualizing the expansion, here's our generic pattern. A plus B to the 4th is 4 choose 0, A to the 4th, B to the nothing. 4 choose 1, 4 choose 2, 4 choose 3, 4 choose 4. A to the 4th all the way down to A to the 0. B to the 0 all the way up to B to the 4th. So let's evaluate. We're going to do B, x plus 3 all to the 4th power. Can I suggest to you, Luke, the most common mistake here are sloppy transcription ones. So you're going to see I'm going to be very meticulous. I am always going to say that's A, that's B. I'll write them above there. Kim, what's my exponent? So it's going to be A to the 4th space, A cubed space, A squared space, A to the 1 space, A to the 0, but I'm not going to write the 0. And it's going to be B to the nothing, B to the 1, B squared, B cubed, B to the 4th. Leslie, see how I put the variables there? What's my exponent, Leslie? So choose 4, choose 0 all the way through to 4, choose 4. And I'll put plus signs in between each term because it helps me keep them separate. By the way, I think that's what it says right at the top of the page, is it not? It's the same expansion, yes? Now let's customize it. What is 4, choose 0? This is 1. What is A? No, A is not A. And my question, what's A? X plus, what is 4, choose 1? 4, what is A? X, what is B to the 1? By the way, can you see what the coefficient's really going to be on my next line? See the 12? Plus, what is 4, choose 2? That I don't know. Is it 6? I can't remember it. Is it 6? 6, x squared, 3 squared. Blaine, can you see on my next line, my coefficient is really going to be 9 times 6, 54, I think? Plus, what is 4, choose 3? 4, x to the 1, B to the 3rd. 3 to the 3rd, got you? Plus, what is 4, choose 4? That's also 1, save yourself some typing. 3 to the 4th. And now I would go to my calculator again and tidy it up. Final answer, 1 times x to the 4th, I think I would just write as x to the 4th. Plus, what does this simplify to? What's 3 to the 1? Times 4, 12x cubed, right? Plus, what's 3 squared? 9 times 6, 54x squared. Plus, what's 3 cubed? I think it's 27. 27 times 4, and if you've got to go to your calculator, that's fine. I think 27 times 4, the 27 times 5, sorry, 25 times 4 is 108, 108x plus, what's 3 to the 4th? 81 times 1, there is the complete binomial expansion of question B. We're also going to cross out question C, by the way. So question B, there you go. Far faster than foiling it out. And I'm going to say more systematic. But can you see there's plenty of room to make sloppy mistakes? I really encourage you. Be careful. Cross out E, not D, cross out E, cross out F. And then in E, instead of a 4 right there, make it a 6. How many terms are they going to be? 7, 7, right? Always one more than the exponent. Because what do we start counting at, Martin? Not at 1. What do we start counting at? 0. Thank you for making that huge effort. Oh, and you know what I'm going to do? I'm going to put a little letter A. A is 2x. What's B, not 3? I got to be fussy. What's B? Negative 3, that means there's going to be some of my terms might end up being negative at the very, very end. What's my exponent set? A to the 6 all the way through A to the, here we go. A to the 6th, A to the 5th, A to the 4th, A cubed, A squared, A, nothing. Charles, what's my exponent? So B to the 0 all the way on up. So no B, B to the 1, B squared, B cubed, B to the 4th, B to the 5th, B to the 6th. Luke, what's my exponent? Choose. 6 choose 0. Don't forget to start with the 0. 6 choose 1, 6 choose 2, 6 choose 3, 6 choose 4, 6 choose 5, 6 choose 6. And I'll start out by putting plus signs in between everything. Later on, I may get a negative kicking around. I don't know. This pattern, this formula, not on your formula sheet. But I don't think it's too bad a one to have to memorize. It's pretty systematic. You know what? I'm going to write the chooses one more time. I'm going to see if I can do all of the calculator stuff in one fell swoop at the very, very end. So I'm going to write 6 choose 0. But what's A according to my question? I'm going to put a 2x to the 6th plus 6 choose 1. A is 2x to the 5th. What's B? Negative 3 to the 1 plus 6 choose 2, 2x to the 4th. Negative 3 squared plus 6 choose 3, 2x to the 3rd. Negative 3 to the 3rd. Plus 6 choose 4, 2x squared. Negative 3 to the 4th. Plus 6 choose 5, 2x to the 1. And yeah, I'll usually put the 1 there just to keep the patterns so I can spot stuff. Negative 3 to the 5th. Plus, running off my page, 6 choose 0. Negative 3 to the 6th. And I'll get my calculator out. Can you see what my coefficient's going to be? It's going to be 6 choose 0 times 2 to what power? What is 6 choose 0? What's anything choose 0? Yeah, my coefficient for the first term is going to be 2 to the 6th. 64. What are my variables going to be? Can you see it? What are my variables going to be? X to the 6th. Someone said x to the 6th, right? Right? OK, but is that OK? Let's do the next one. 6 choose 1 times 2 to the 5th times negative 3 to the 1. Now, why did you type that, Mr. Dewick? Some of this we could do in our head. I agree, but I think it's going to be the same pattern for the rest of the questions. So I'm going to start going second function, enter, and just changing numbers. The second coefficient is 6 choose 1, 2 to the 5th, negative 3 to the 1. The second coefficient is minus 576. What am I ver- Sorry? X to the 5th. And Martin, here's why I typed this whole thing. If I go second function, enter, the next one wants me to go 6 choose 2, 2 to the 4, 3 to the, sorry, negative 3 to the. By the way, you have to put the negative 3 in brackets. If you just put the exponent on the negative 3 without the brackets, you will not get the right answer. You know what my next coefficient is? 2160. Positive or negative? Positive. X to the 4th. Next one. Second function, enter. The next one is going to be 6 choose 3, 2 to the 3rd, negative 3 to the 3rd, negative 4320, minus 4,320 X to the 3rd. Next one. So I've done the third one. This one. Second function, enter. 6 choose 4, 2 to the 2nd, negative 3 to the 4th, 4860. Positive. X to the square. Next one. 6 choose 5, 2 to the 1, negative 3 to the 5th, negative 2,916. Variable looks like just an X. The last term. You know what? This one I will take the shortcut. What's 6 choose 0? Sorry, I wrote 6 choose 0. You know what I should have written over here? 6 choose 6. You know why I wrote 6 choose 0? Because it's the same answer at 6 choose 6. I was already thinking that. What is 6 choose 6? Also times negative 3 to the 6. The last term is going to be bracket, negative 3 to the 6th power, positive. Oh, Mr. Dewick, he's in the green, 729. That is the binomial expansion of 2X minus 3 to the 4th to the 6th power. Doing that by hand, you're not doing it in three lines. I guarantee it. Yes. Yes. Huge. But I can't tell you for about a month. But actually very huge. We're going to use a variation of this to figure out, for example, the odds of passing a 40 question multiple choice test by guessing. We've really done the odds of getting perfect, but what we were not able to do was do the odds of, say it was 40 questions, of getting 20 or 21 or 22 or 23 or 24 or 25, et cetera, et cetera. What does or mean? Add. We could do them all separately and add all 21 examples, but maybe there's a shortcut. And it's going to use a variation of this. Totally. Fugely useful. On your test, I'm not going to ask you to do a seven term like this. I'm going to give you a big exponent, probably like an eight or a nine. I'm going to say find the first four terms. But it was worth doing a whole term once. That's not all I'm going to ask you. On the test, I'm going to give you a binomial to the, oh, Kellen, let's say, ninth power. And I'll say find the sixth term only. You can do it by finding the previous five. But it would be wonderful if there was a shortcut, Martin, that let us jump right to one term. And there is. Turn the page. The problem here is, do you remember on the previous page when we wrote the pattern, I said that it was n choose term number minus one? It gets very confusing. What I'm going to do is I'm going to give you the term K plus one formula. Over here, this is the term K plus one formula. And it is, write this down, n choose K, a to the K, b to the n minus K. Now, that looks horrendously confusing. It's actually about as plug and chug as the quadratic equation from last year. Once you know what everything means, here's what this is saying. Suppose you want the eighth term. Don't write this down, just watch. I want term eight. If I want an eight right there, what does K have to be? Seven. Over here, I would write K is seven. And it would be whatever your exponent is choose seven times a to the seventh, b to whatever your exponent is minus seven. And you plug in the appropriate numbers. Why do we have a K plus one here? Do you remember on the previous page it was term minus one? We could either have had a minus one there, a minus one there, and a minus one there, or we could just adjust this little index and have a normal looking equation. I still haven't showed you how this is used, and I haven't cleared this up. Here's what I mean. By the way, this equation, who has their formula sheet in front of him? Anybody? I see right there, Alyssa. Or is that physics? This is on your formula sheet. It's on your formula sheet, Alex, right next to the choose. Is it not? So you don't have to memorize this, but this is how you find a single term if you want. Am I wrong? Sorry, say it again. Read me what it says on the sheet. N choose K. Thank you. Fix this. N minus K, b to the K. I've memorized this one either. Sorry. Here's a question I'm going to give you on your test. Find the fourth term of that. Now, you could write the previous three, but instead, we're going to try and use this monstrosity here. And the first way to use this monstrosity is to write it out. Write down all of us, please. Term K plus 1 equals N choose K, A to the N minus K, b to the K. What's A in this binomial? X. What's B in this binomial? Negative three. What's N in this binomial? I'll give you a hint. It's the exponent. What's K? This is the only tricky part. Look up. I want term four. That's what the question says, yes? What does K have to be to get a four there? Three. That's the little adjustment we have to make. And that comes out of the fact that we start counting at zero. Once I've made that list, physics 12, yes, we're deficking. Once I've made that list, it's going to be N. What's N? Nine. Choose what's K. Three. What's A? X to the what's N minus K? More specific. Six. What's B? Negative three. What's K? It's going to be this simplified. Let's do the numbers first. It seems to me that the numbers are going to be nine. Choose three times. What number's in front of the X here? It's invisible. Yeah, one to the sixth. Am I going to bother doing that? No, but if it was a coefficient, I'd do that times negative three to the third. You know what the coefficient of the fourth term is? Negative 2,268. What about the variables? X to the sixth. That's how this works. It looks ugly, but Stephanie, if you list things carefully and adjust the K, that's the key, because if this is multiple choice, do you think I'd have one with K equals four? Yes. If you adjust the K, then it really is plug and chug. And you don't have to memorize this formula. In fact, you saw, I don't even have it memorized. It's on your formula sheet. So it says write the first four terms of the binomial expansion of that bad boy. Instead, what I'm going to say is this. Write term eight. Find the eighth term. I'm going to freeze the screen, see if you can try this on your own. Write the equation, make a list, and then plug stuff in. I find the kids that try plugging stuff in directly into the equation without writing it out make dumb mistakes. Don't know why, so I've learned to take that one extra second. So write the TK plus. Oh, what is it? I've frozen the screen. Yay. If you're stuck, there's my list. Yes, I think. And Martin, partly in answer to your previous question when you asked where he uses, sometimes it's just nearly cool to find shortcuts like this. Like the math third with me does say, got to say, that sure better beats doing it by hand. But hold the original question. Trust me, about a month or so, will actually be, it's something called the binomial probability distribution function and hugely useful. And not just even in math, would you believe, I actually used that once in a while in real life to figure stuff out. Football, probably not so much, but you've heard the whole money ball thing in baseball where they really started taking stats to an umpteenth level. A lot of that is this kind of thing here. Fascinating, keep your attention here. Can you turn it, can we get that okay? 101,376 x to the fifth, y to the seventh. What did you use for your, do you have the same data over here as me? And you had this, you had this as well. So I went 12 choose seven, right? And then times two to the seventh. See it, right? Right? Is that okay? Makes sense? No, no, not kind of. It's gotta be, because the two is part of the B which is to the seventh power. So there's gonna be a, in terms of numbers, it's gonna be that's a number and that's a two to the seventh and they're gonna get multiplied. And then your variables, the exponents go on top. Right? Turn on your workbooks, please. Gonna turn back to lesson six. Page, except the homework is four, 414. Well, yeah, the workbook combines Pascal's and the binomial, okay? So homework questions that I'm going to assign and I'll do these in a different color from the previous ones, but I think number one all except you can cross out D. Page 414, Steph, page 414. Page 4A we're gonna do right now together. If this expansion has 18 terms, what's the exponent M? Not 18, 17, because you always end up with one extra term because you start counting from zero. So what I am gonna say is try this one. If you know there's 26 terms, what does that have to be? That's the kind of mini curve ball that we'll throw at you for the term question. 7A, 7B, excuse me, 7C, 7E, so I skipped D, 8. Now what I haven't talked about is going backwards and that's gonna be a half lesson next class. So next class we're gonna ask ourselves, for example, what if they tell me the term, can I go backwards and figure out what that had to be? Or what if they tell me one of the exponents can I figure out when that occurred? 12th, and there is your binomial expansion theorem and I also have a take home quiz, yes? Did I give you the take home quiz? Gotta give you a take home quiz right now.