 Okay, so I think what I'm going to do is blaze through the whole unit in about 15 minutes key ideas looking at questions from the review that I think are good. And then I'll try and highlight specific ones that I think are difficult. So I have written here difference between permutation combination Kyle don't forget to let me to come back. Don't let me forget to come back to that. But we started out with the first thing, which was fundamental, fundamental any type counting principle. Where are we going to use the fundamental counting principle stuff like question number four from the review. So let me actually open up the combinatorics review. Number four is a good example of the fundamental counting principle. So number four says there are 45 multiple choice questions and sorry for the smudging this of it on an exam with four possible answers for each question. How many different ways are there to complete the test. So there's 45 different choices to be made. How many options do I have for each event or each choice for this is going to end up being four to the 45th because what we would have done is we would have drawn 45 blanks and we would have said four choices for choices for choices for choices. We would have done that 45 times and then we would have said oh 45 four is in a row. That's four to the 45th. That's an example of the fundamental counting principle and get fancy and bring in any permutations or choose or anything like that. Okay, and I'm going to close this version because I have one that's less malaria I think but I mean they have to open it in word. Other stuff we did in the fundamental counting principle. I can find another good example here. While my word file is slowly opening. Here we go. Page with have stuff up here. Digit and number questions are good examples of the fundamental counting principle. So number 11 is going to start out similar to the fundamental counting principle. But we're also because some of the numbers in here repeat, we're going to have to use some of the permutation skills that we came that we brought in later. I'm going to come back to number 11, as well as number 12. Sorry. Let me see if I can find another good fundamental counting principle question. Sure. Number 23 would be a good example of another fundamental counting principle question. And this is a classic question. License plates are easy for us to make nearly interesting questions. It says a car license plate consists of seven characters. One, two, three, four, five, six, seven. The first character can be any of the letters from A to F, but no letter can be repeated. So A, B, C, D, E, F, that's six letters, but I can't repeat. So there's five left, then four left using our fundamental counting principle. The next three letters can be any of the digits from one to nine, but no digit can be repeated. So how many choices what I have right here? Nine, then seven. And the last character can be anything x, y, or z, three. And they even give me an example, B, F, A, six, four, eight, y. How many different outcomes are there? By the way, I could go six times five times four times nine times eight times seven times three. I do notice this is actually nine, eight, seven, six, five, four, three. So by a fluke, I could do this as nine factorial over two factorial, which is a lot less typing. Or this just happens to be nine P seven. That's a fluke. That's not the rule for doing this question. This question is a fundamental counting principle. I'm sure whoever made this up, though, deliberately tried to get a six, a five, a four, a nine, a seven, an eight, and a three to appear in the question somewhere. Anyways, what's the answer? B. So those are examples of the fundamental counting principle. Then we broaden that into permutations. Oh, by the way, another question under the fundamental counting principle that I love is a question involving digits. And if I ask you, how many four digit numbers? What can't you start with? Zero. Please don't forget that. And maybe I'll ask you for an odd number or even number or less than 300 or whatever. We've done all sorts of variations on that. Permutations. Now this is an ordered arrangement. And this goes back to Kyle's question then. Kyle, one of the ways that I can tell it's a permutation is if they use the word arrangement and they don't use the word unordered in front of the word. Arrangement. Show you what I mean. If I go back to the review, back to number one, let's see. So in a permutation, order matters. I looked at number two at first and I asked, is that a permutation or a combination? And I thought, if I'm selecting fruit, do I care which fruit I selected first? Or do I just want to know if I have the fruit or if I don't have the fruit? I think number two, although I have to think about it, is a combination, not a permutation. Number three, a combination lock is misnamed. A combination lock should actually be called a permutation lock because I'm telling you in your locks the order matters. You all have three numbers on your locks, but you have to do them in the correct order. So a lock, a combination lock, should be called a permutation lock. The order matters there. Oh, number seven is a great example of a fundamental counting principle question. For some reason, I missed that. When I look at number seven, we have juice, toast, eggs, and beverages. How many different breakfast specials are possible? Three times two times three times three. Okay. Number eight, Kyle, I see the word, how many different ways could the colors be what? I'm leaning towards a permutation. And now I have to think and say, does the order make a difference? Yeah, you know what? She's using the same colors over and over and over. She wants to know how many ways she can mix up the colors. So number eight here, I think would be a permutation. Now permutations came out of the fundamental counting principle. We could go, there are seven colors, one, two, three, four, five, six, seven. And we could say we have, because she can't remember the correct order, so order does make a difference, seven, six, five, four, three, two, one, which is how we started out by doing this. And then we said, you know what? There's a great shortcut for writing that. How can we write that? Seven factorial. And I think seven factorial is 5,040, I think. Okay. So there is an example of a permutation. Looking for another permutation question. Number 22, number 22 has the word arrangements, but I can't use this as a straight permutation because there's letters repeating. And that was our third topic. So I'm going to come back to permutations with letters repeating in just a second. But number 24 is a great permutation question. So it says a soccer coach must choose three out of 10 players to kick tie-breaking penalty shots. Assuming the coach must also designate the order. Oh, order matters. How many different arrangements are there? I think I would call the soccer players A, B, C, D, E, F, G, H, I, and J. And it's a word. You can do it 10 choices, 9 choices, 8 choices. Or the shortcut we said for this was the notation from 10 objects permutate three of them. And you'll recall on your formula sheet, this is actually 10 factorial over 10 minus 3, 7 factorial, which is why this gives you the 10, 9, and the 8 because the 7, 6, 5, 4, 3, 2, 1 would cancel out. This was by far the fastest to type. This was the easiest to do in your head. 9 times 8 is 72 times 10. Hey, the answer is 720. Oh, but in this case, they want the factorial notation. They want this, and you need to be able to recognize whether I want the answer. You'll look at the answers and figure it out. Sometimes they'll want the actual simplified answer. Sometimes they'll want it in factorial notation, but the answer here is A. Then we did permutations with repeating letters. Permutations with repeating letters. Go back to the beginning of this review. Let's find some with repeating letters. By the way, number 5, North American area codes. I'm pretty sure that's going to be a fundamental counting principle, drawing out the blanks and filling them in carefully. Because it says the first digit could not be a blah, blah, the second digit, the third digit. Yeah, you know what? That's going to be how many choices for each one. Excuse me. Here's an example number 12 of one with repeating terms. How many ways can four colas, three iced teas, and three orange juices? I said, this looks to me like four colas, three iced teas, and three orange juices. That looks to me like a 10-letter word. And if we're trying to see the number of ways that we can mix that up, how would we do this one? I said it's going to be 10 factorial because there's 10 different beverages over what? Four, three, three. Are you all okay in typing that into your calculator? What do you have to remember for the denominator? Brackets, although I'll be honest, I wouldn't type four factorial, three factorial, three factorial. I would type bracket 24 times 6 times 6. That's a lie. I would type 24 times 36 because I know that 6 times 6 is 36 because it's a lot of typing and I start to take shortcuts. But you can do the whole smear if you want to. What's the answer? Sorry? What's the answer? How do I know this is not Kyle a choose question? Because if you get an orange that's different from you getting a cola is different from you getting an iced tea. Right? So what your choice is makes a difference. Let's do another permutation with a repetition. There was a nice one here that had a little bit of both in it. Where was it? Oh yeah, number 11. Number 11. So I've told you I'm going to ask you some kind of a digit question. This is a bit of a trickier one. It says how many six-digit numbers. This is a combination of fundamental counting principle and permutations. Since they said six digits, I drew six blanks. We have to be greater than 800,000. What number has to be here if I'm going to be greater than 800,000? So I said, you've got to put an eight there. You only got one eight in the bucket. Now I was going to then go five, four, three, two, one. But I said, ah, problem. Some letters are repeating. See it? I got three fives and I got two ones. I said, ah, I'm not going to draw blanks. This is going to be a word. This is going to be a word with five letters in total. Three of them repeating. Two of them repeating. Right? Five factorial is 120. Six times two is 12. I think it's 120 divided by 12. I think the answer is 10. I hope I'm right. I think I am. Yeah. No, no, no, no, no. We put the eight there already at the front. Ah, good question. I think we have to assume that they can't repeat because I think it's saying make them from this group. Now this is from the exam specifications. This question probably wouldn't have actually made it onto an exam. Someone who was overseeing or monitoring teachers pre-write these and were very fussy. Someone would have said, hey, we need to tell them, comma, assuming letters cannot be used more than once. Good point. The real answer is, because I looked it up and the answer was A. Okay. So permutations with repetitions. I'm going to definitely give you something like apple pie, like number 22. But I think all of you are okay on that, right? You've done letters and you're fairly comfy with them. The other thing that we added to our permutations with repetitions is we said this is where we can first solve regular pathway problems. Let me find a regular pathway problem. 56, because I don't think I assigned 56. This is a regular pathway problem, because it's three separate rectangles. This first rectangle would be right, right, right, right down. This first rectangle would be five factorial over four factorial, one factorial. I probably wouldn't type one factorial, but I'll write it there just so I can double check that the bottom ends to five. Without me labeling it, what would the second rectangle be? Two factorial all over one factorial, one factorial, right? Oh, wrong button. And what would the third rectangle be? One, two, three, four, four factorial over two factorial, two factorial. Now, there was a wonderful coincidence for regular pathway problems if we noticed that these two added up to the top, that these two added up to the top, that these two added up to the top. There was a shorter way to write this. This is actually five choose four. This is actually two choose one. This is actually four choose two, which gave us a quicker way to type it. I always start out by solving it this way, but oh, if the bottom adds to the top, that is what choose look like. I have no idea what the answer, oh, wait a minute, I do, because five choose four is five. Two choose one is two. Four choose two, I don't know what is four choose two times ten. Just go four choose two on your calculator, hit enter, and I'll multiply it by ten in my head because I can. What is four choose two? Six, sixty. And that led us to, we finished off permutations with repetitions doing this pathways, but that led us to the next lesson, which was combinations. You can, yeah, long, but yes, combinations, unordered arrangements. Now they'll rarely use the word arrangement to refer to a combination. They'll often just say chosen. If they do say the word arrangement, they'll put the word unordered in front of it. Order does not matter. Card questions are a great example. Committee questions with no president, vice, president, et cetera are a great example. So let's try a few. From your review, let's jump to the written section. Right near the end. How about number 14? A theater company of 13 actors consists of eight men and five women. How many different ways are there to choose from the theater company a group of seven with exactly three men? So that's A. I said for choose questions, my best strategy when there's more than one group is a bucket. In this case, we're going to have men and women. How many men are their grand total? Eight. How many women are there in the bucket? Five. How many men are we required to choose? Three. How many women are we required to choose then? You'll have to do some thinking. How many? Four. But not four. I'd like it with some authority. How many? I expect you to go seven, take away three in your head and get four. Seven in the group. Three guys, then four girls. What's the equation? Eight, choose three and, and means multiply, three guys and four girls. Three guys and four girls. What's the answer? Sorry? 280? 280. Okay. And then I guarantee you, I'm going to give you a case question where I use at most or at least, and you need to list the cases. Pretty much like B here. So, same bucket. And on a test, I would probably redraw it because it takes all of two seconds. Now we want at least four women. And we're picking a group of six. What does at least four women mean? What cases does that include? Four, or, or what? Three is at least four women. Okay, we need to know our English here. Five, or six. What does or mean? Add and means multiply. That's why I said and here. Or means out. So if I have four women, how many guys am I picking? Careful, it's not seven in the group this time. How many are in the group this time? Six, yes. So how many guys am I picking? Two. The first term is going to be eight, choose two, five, choose four. And then I wouldn't redraw the bucket, but what I probably would do, Blaine, is cross that out and say five, cross that out and say one, eight, choose one, five, choose five. Or, oh, can I have six women? Why can't I have a group with six women in it? Okay, and my bucket kind of, I mean, this is usually where I've ended. I can't go five, choose six. So I'm going to scribble that out, right? Jordan, does that make sense? Oh, and by the way, eight, choose one is eight. Five, choose five is one. Hey, that's an eight. So I really just got to type this one. What is it? Anyone, anyone, anyone. 40, 140. Final answer, 148. Now let's see if I can find one that has both a permutation and a combination in it. And I think there is one on the written. Take a look at number 13 for a second. Look at eight and ask yourself whether you think that's a permutation or a combination. Kyle, why? Well, first of all, they use the word chose, which is what I've almost always said when I use combination. I don't think order matters. In fact, there's seven different chairs. I think it's seven, choose three would be my answer for that. Now compare that with B. What word did they use in B? Arranged, I'm thinking permutation. In fact, I think this is a word. This word has two B's for blues. Three Y's for yellows. One R and one G. And I would do seven factorial over two factorial, three factorial. I'd mix up that word. Okay, so the answer to your question at the beginning, Kyle, is I can't give you a great definition. If you do a bunch of these and there's a gut instinct, and it's really hard to put my finger on it, I can give you a few hints. Arranged! Most of the time I sit and think about it and would it make a difference? Some are obvious. Like number 12. Okay. Number 12, seven boys, five girls. I think the first question is a choose, and the second question is a mixture. Why do I think the second question has a permutation aspect to it? They're mentioning positions. So let's try 12. Try A on your own. I'll do it up here. I'll freeze the screen, although the video people at home won't get the frozen screen. They'll get to see me working in cheap. I got that. 70? Yes? No? Yes? Yeah. One boy. I guess three girls. B. Okay. I think now we're going to combine some of our fundamental counting principle permutation along with choose. Let's see. We must have a female president. So here is our female president. How many choices do I have for my female president? How many girls are in the group? I got five choices. And a male vice president. How many choices do I have for my male vice president? Seven. And, and then they say two others from the remaining. Does order matter for those remaining two others? Does gender matter for those remaining two others? So we started out with 12 students. We've picked two of them. I think now it's going to be from the 10 remaining. Just choose two. I think that's how I would do that one. Does that make sense? That's considered a fairly tricky question, by the way. You can go five choose one, seven choose one, because as it turns out from the fundamental counting principle, five choose one. Yeah, but five choose one is five. We're saying I fall back on the fundamental counting principle whenever I can. What is the answer, by the way? One five seven five? Yes? Anybody else? Yes? No? Yes? Okay. All right. Then we moved so that is that okay for permutations and combines and combinations and it's going to be a card question. I haven't done one, but I think you guys are okay on the card questions. Just make sure you know the difference between how many red cards, how many black cards, how many face cards and there's 52 cards in a deck. You do need to come know that. Then we moved on to I think I'm on item number five now. Pascal's triangle. And there's going to be two main types of Pascal's triangle questions. The first one goes like this. What is the 13th term from the 27th row of Pascal's triangle? What is the 13th term from the 27th row? How do I find it? Now, sorry. This is the most common mistake. Try typing 12 choose 26 on your calculator. What's going to happen? Try it. Brett, what you're saying is from 12 objects pick 26 of them. Thank you. The only problem is in English grammar I do have to say it this way. Right? That's the way that sentence has to be structured. So yeah, 26 choose 12. Which is what? Big, I'm assuming. Read me the digits. 9, 6, 5, 7, 7. Oh, oh. And I've slowly gotten bigger because people kept drawing Pascal's triangle. No, it's minus one, minus one. Second most common mistake because that one your calculator would have barfed it and you would have figured it out anyways. Second most common mistake, people would go 27, 2, 13. Nope. It's one less, one less. Why is it one less, one less? Because it started counting from zero. It's irregular pathways. Let's see if I can find one. Sure, 51. 51. Is 51 your 1, 2, 1, 1 press? And he changed the numbers. That's what it used to be. In theory, but he changed it one year because I guess people had scattered it too much. Excuse me. This is an irregular because there are shapes cut out of it. Are you folks okay on these ones? I can, totally. I would love to be able to rotate it 45 degrees. Sadly, I cannot. However, I would start out by going 1. Do you know what am I going to put here? Next, with authority. Yep. Yep. Yep. Mm-hmm. 10 plus what? What's right there? So, what's going to go down then? What's going to go down then? 10 plus what? Zero. What's going to go here? 10. Yep. 25 you said? I think, yes. 10 plus what? In fact, I think that 10 is going to be dropping down. Anyways, we'll get that. Oh, 6 plus what? 51 you said? 29 you said? And 51 and 29 is, low and behold, 80. So, I'll be really blatant. On your test, you're going to have a regular pathway on the multiple choice. So, like the one that I did earlier with the chooses, that was three separate rectangles. And then on the written, you're going to have an irregular one like this with shapes cut out, or it's not quite a rectangle or whatever. Okay? So, there are a few more of those for you to practice on your big review. 41 is another good example. 41 is a bit weird. Why don't I try 41 as my last one? Did I assign 41? Okay, well, we're going to do it anyways. Who would like to try this one? Sorry? You? Okay, ready? Here we go. One. All the way down to there, and I'm going to do that for a second because I better do this one. What about here? Ready to do that guy? Nothing there. See it? That's why I hesitated for a second. I was nothing there. I'm missing a line, so I'll just pretend nothing and keep going across. Here, one plus four is five. Right? Just because that line is still flowing into that path, Krikov's Laws is another unique application actually of Pascal's Triangle as the currents break up and join it. It's the currents flowing in just like we've been doing. Don't panic. If they give you something weird, pause. By the way, I find for nasty ones it helpful for me, Jordan, literally with my fingers to do that, and then I can figure out what's flowing into there. And it does an empty part. I put my finger there and there's no line, so I'll give you zero. Whatever works for you, but that was my figuring this out routine. So there was the variations and perturbations and permutations of Pascal's Triangle. The last thing we did, which I think is item six, was the binomial expansion theorem. There was questions about this as well. So this is what I'll be spending whatever, minutes on. And I'll give you some hints as to what you're going to see on the test. Now, you need to know the pattern. In other words, if I say write the first four terms and simplify of better make sure I don't accidentally make up one on the test, 5x minus 2 y squared to the sixth. Can you do this? If I pause right now, try it. Can I do that? I'm going to pause the video even this time. You can try this at home. Those of you who are following along yourselves will be right back after a word from our sponsors. Those of you scoring at home were back. I did that while I had the video paused. I think the next thing I would do is very carefully do my substitutions and my chooses. So 6 choose 0 is 1. 6 choose 1 is 6. 6 choose 2 I found was 15. 6 choose 3 I found was 20. And I carefully with brackets substituted everything in. And I can now go to my final answer, I think, I hope. Most common mistake kids go, oh, the first term is 5x to the sixth. It's not because that 5 is to the sixth power as well. It's going to be 5 to the sixth. It's going to be 15,625 x to the sixth. 15,625 x to the sixth. The second term is going to have an x to the fifth and the y squared. But it's going to have a coefficient. There's going to be a 6. Times there's going to be a 5 to the fifth. Times is this too negative? Put it in brackets just to be safe. Negative 2 to the 1. The second term is going to be minus 37,500 x to the fifth y squared. You know what the third term is going to have? A 15 times 5 to the fourth times a negative 2 squared. It's going to be a positive 37,500. That's kind of nice. Positive 37,500 What are my variables going to be? x to the fourth y to the fourth. See it? The last term, by the way, can you see the last term is going to be negative? Because it's a negative to an odd power. The last term is going to be 20 times 5 to the third times bracket. Negative 2 closed bracket to the third minus 20,000 x cubed y to, oh I missed the squared over here. I'm going to do it because it is going to be y to the sixth. This one, because there's an exponent inside the exponent, I can't use my normal check of, do they add to six, add to six, add to six, add to six. Although it worked up here when I had my a's in place, yes. But sadly, my normal check does not work. So I was going to do one of those on your written. Are you okay in finding a single term using the t-cup t-k plus one equation? Then what you're wondering about is going backwards. That was one of the last questions that someone asked. The t-k plus one thing? I will in the going backwards anyways. I think that will suffice. And I'm going to make one up because I want to give you practice. I think I gave you a couple to try on your review. I don't want to use one of those. But to make one up, I'm going to freeze the screen and the video. So here's an example of a going backwards question. One term of the expansion of 2x plus b to the ninth is negative 18,432 x squared. So here's my argument. What I'm going to do is reverse engineer this thing. But first, I need to figure out what the term number this is. And the way to figure this out is by looking at the exponent on the variables. I have an x squared. Now in this particular question a is 2x. I'll deal with the 2 later. But I know that I would have gone a to the ninth in my first term. a to the 8th b a to the 7th b squared unless I usually when I'm trying to find when something occurs, I write out 3 because if I can't figure the pattern out from 3 things, I'm in trouble. What's a ignore the 2 what's a just so I can figure out how the x's are behaving? x. I think I had 9 x's in my first term. You know how many x's I had in my second term? 8. You know how many x's I had in my third term? What's the pattern? We seem to be going down by 1's. Now if there was another exponent there by the way I wouldn't be going down by 1's but in this case I'm going down by 1's which means I can count my way to figure out when a squared would appear. Get my fingers. Ready? What did we start out with? x to the x. Then 6 5 4 3. You know when the squared is going to appear? Term number 8. Aha! I want term 8. Now here's what that means. I'm going to make my list over here. A is 2x B is B What's n in this question? 9 What's K? 7 And when I fill all this in then I'll get 9 choose 7 2x n-k squared B to the 7th. I'm going to tidy up the right side as much as I can. I'm going to go 9 choose 7 times 2 squared. What is 9 choose 7 times 2 squared times 4? Sorry? 144? So I have this 144 x squared B to the 7th. Now initially I wrote term 8. Now I'm actually going to write term 8. What is term 8 equal to? What do they tell me the term worked out to? Negative 14,432 x squared. What happens to my x squareds? Thankfully they cancel. And how would I get the B to the 7th by itself? What would I do with this 144? Divide. I'm going to get if I divide by 144 divide by 144 I'm going to get negative 128 he says doing some math in his head because it's multiplying how would I move myself? Divide. We cancel out the x I know but it's still times times times. Plus sign suddenly magically appear. Oh now that gets me B to the 7th and it will always be some kind of an exponent here when you use this method. How would I get rid of a 7th 7th root which I can do on my calculator. B is going to be the 7th root of negative 128 which I'm pretty sure is negative 2. You need to know how to find this on your calculator. So make sure you know how to do the 7th root on your calculator. 7th root of negative 128 is negative 2. Apparently B was negative 2. Apparently my initial binomial was 2x minus 2 to the 9th and you actually may remember I had a negative there and quickly erased it so that it would appear down here because I forgot to get rid of it when I made this example. That's going backwards. I gave you a couple try in your notes in the homework and in terms of the review let's work my way backwards if I'm intended. Good gosh are there none of these in the review? There's lots going forwards. Yo Hang on. Okay there's going to be one of those on the test but I just decided to make it worth fewer marks because I don't see one here on the review. I always thought there was one on there. Well there's two in your homework. So Number 6 That sure though what sorry? Negative 2. Is that okay? Oh clever he's actually writing this tutorial in the back cover of his workbook where the blank pages are. Well Rio showed you. Rio! Well done. Number 6 They said the 10th term the 10th term is x to the 5th. I said okay if the 10th term has an x to the 5th now that x to the 5th came from that x so we would go and then one last one last one last and eventually we did up with 8 of the 5th. So I counted on my fingers. I said I'm either going to be starting with a 15 there if I started with a 15 there the first term would be x to the 15th x to the 14th x to the 13th x to the 12th x to the 11th x to the 10th x to the 9th x to the 8th x to the 7th x to the 6th x to the 5th is the 11th term. What do I want it to be? So the 15 is wrong then I tried 14 that's what I wrote. I counted on my fingers and then I went oh that kind of makes sense because I wanted to go 10 plus 15 but we always start counting from sorry 10 plus 5 but we always start counting from 0. I bet it was the 14th as an exponent right there which gave us the anyways I counted my fingers to figure out when an exponent appears. What a term number is. Write out if you spot the pattern in this case because they were only asked me to find n and I knew I was going down by ones because there was no exponent on the x. I didn't write out 3 and find the pattern. I started with my answers. Is that okay? Now what I have not talked about just realized is yep simplifying factorial expressions as a multiple choice question so something like like something like number 40 okay but it's going to be a little tougher than that one. I think it's going to be more like something like number 34 maybe or something like there's one right near the beginning if I recall Mr. Deweyck like number 6 or 7 or something like that I'm getting closer. Yeah something like number 9 for me yep that's what I did for number 9. The first thing I did not like this n factorial let's do it Mr. Deweyck Don't take the lazy way out because you're tired but I am tired I didn't like this n factorial squared I said you know what I'm going to write that as n factorial n factorial because that's what n factorial squared means and then I said and I'm just going to break this up into too many whoops don't want to cover up the exclamation mark fractions there one fraction but I'm going to simplify this and this so I ended up with I had n minus 2 factorial over n factorial and n plus 1 factorial over n factorial what's bigger the bottom and I started to expand that so I left the top the bottom became n times n minus 1 times hey n minus 2 factorial stop writing there what's bigger what's bigger the top so I said this is n plus 1 and I'm going to stop writing there n factorial can I cancel it's all factored I said I could cross out those two I could cross out those two and it looks like on the top I'm left with an n plus 1 see it and the bottom I'm left with an n and an n minus 1 oh deep so if they give me four terms chunk it up into two and two remembering that they're still multiplied together so you'll see I didn't actually separate the fraction I just drew spaces between them but I kept the same fraction but I still have the same fraction but give yourself a break don't try to keep track of everything that's a tough one where I get what okay so then what I'm doing is I'm saying so I'm going to expand this this times one less but I'll stop when I get the matching term on the top so you okay with this line here you'll get this line here I just basically said those two are going to go together and those two are going to go together n plus 1 times n factorial is what's on the bottom here so that's why I quit right there an n minus 1 if I kept going but I already got the term that I wanted to cancel here right the same reason I stopped at n minus 2 here was because it was what I wanted to get rid of on the top the same reason I stopped at n here what I wanted to get rid of on the bottom was an n factorial but it would be n minus 1 factorial n minus 2 factorial n minus but this is n minus 1 factorial n minus they would all cancel yeah yeah that is 99% of the test I think you should find about 60% of the questions pretty plug and chug about 20% of the questions are going to be a bit tougher like irregular pathway or you got to do some thinking a little bit on the multiple choice there's one EDM tricky one and then the last two written questions well I think actually the second last one most kids will get at least two out of three a lot of them will get three out of three and the very last one was going backwards and I just decided I'm going to make it work less marks than I have and I'll make that correction when I hand the tests out I didn't realize there wasn't one on the big review which means it's unfair to make it work a lot of marks but that question will be worth very many marks that's your guess so I'm going to hit stop on the recording does anybody want me to print them a copy of this