 So what's tricky about this unit? Well, one of the things that I think Calvin was asking about this is deciding when it's a permutation and when it's a combination. So a permutation is when the order makes a difference, when first is different from second, is different from third, or the first person picked has a different condition, the second person picked has a different condition, then it's going to be some kind of a permutation. But I'll be honest, if I'm not sure, I fall back to drawing blank lines and going on to my country principle. Combination, choose, when do I use it? Cards, cards. For a committee where they talk about genders, boys and girls, and they don't mention specific positions. Lots of other stuff that we haven't really got into. But really, that's one of the tricky things is asking permutation or combination. I'll be honest, on the test, remember that the test is going to be roughly in the same order that I taught you. Which means number one is not going to be a combination, because I hadn't taught you combinations at the very, very beginning. In fact, I'll tell you right now, number one and number two are going to be fundamental counting principle, probably. I'll even be more specific. One of the questions somewhere will be asking you how many, I'm going to make this up, six digit odd numbers, how many, five, or I'll give you a group of numbers and say how many digits can you make up from, blah, blah, blah. And it'll either be with repetition or without repetition, and I'll be make sure that the question is clear. Then almost certainly there's going to be a word with all the letters. There's going to be a word with all the letters, but some of the letters will repeat. So there's going to be a word like camp, where no letters repeat. And then there's going to be a word like Mississippi, where some letters repeat. Oh, and then there's going to be a word with some of the letters. And those are all permutations, because you're rearranging. Okay. The word arrangement is a trigger word for a permutation, unless it says unordered arrangement, that's a combination, but the word arrangement is also one that I would use as a trigger phrase. Okay, so let's try a few of these multiple choice one and a half mark each for some reason, just like the provincial exam. Why do they do that? I don't know. At the cafeteria at PMSS, Chef Dewey has created four feature meals, three drink choices and four dessert choices. I think it's going to be four times three times four. By the way, I noticed some of you on the quiz did this. You said meals, there's four of them, drinks, there's three of them, dessert, there's three of them, and you went four choose one, three choose one, three choose one, which I guess also works because of that is four times three times four, and you're choosing, you know, you know what, it's way easier, I think, just to do fundamental counting principle for those first ones. But I did notice that when I was going through the quiz and some of you did that. How many different arrangements can be made of using, oh, correct answer, what is four times three times four, good question, Mr. Deweyck, four times three, B, is that right, how many different arrangements, so Kelvin, when they talk about arrangements, that's a permutation, can be made using all of the letters of the word Toronto, Ontario, one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, there's 14 letters. How many T's, three, how many O's, five, how many R's, two, I already did O's, how many N's, two, I already did T's, I already did O's, I already did O's, I already did N's, I already did T's, one A, I already did R's, it's going to be that. Jen, how will I type this into my calculator? I'm going to go 14 factorial divided by, what am I going to have to remember to hit? Darn right. I know three factorial is six, five factorial is five times four times, 120 times, two factorial is plain old, two, two, I'm just going to put a four there, and you can put the factorial in as well, I'm just lazy, that's so much typing. That, D, is that right? Number three. What value is equivalent by definition to 1000p4? So we said NPR on your formula sheet was N factorial all over N minus R factorial, so 1000p4 would be 1000 factorial all over 1000 take away for 996 factorial. It would be 1000 times 999 times 998 times 997, it would be that. Oh, B, yes, I almost said none of these and then I went, oh wait a minute, they gave it to me in that notation, almost fooled me. How many different three letter arrangements of the word symbol are there? Are all the letters different? Check. So I'm going to go six, it's an arrangement, permutate three, because they want to mix up the letters. And six p3 is 120. Oh, none of these. On your test, Elizabeth, I won't have none of these as an option, but the person who I borrowed this from, this is a teacher I think from North Van, otherwise it was a good test covering everything, they like to do the none of the, okay, that's fine. Provincial does not do none of the above either. Yeah, we've been very clear on that. In fact, in complete honesty, I think that's a lousy cop out of an answer. By the way, also on the provincial, it's four options, not five. Bonus. The letters of barrier are arranged all the time. How many different arrangements are there with the first letter R? Why did I write bonus? I wrote bonus because I was, I made a mistake, I think the last time I did this, give me one second here. Desktop. I'm actually going to look at my old answer key to double check here. Combinations. Make sure I open the rate one. Combinations review. Combinations review. Quiz two. Semester two, take home test answers. Which is not the same as semester two test, right? Right. Semester two, take home test answers. Better pause the, freeze the screen just in case. Screen frozen? Of course, my video recording isn't. Good. So, what did I say? My first thinking was, pick an R. How many ways do I have to pick an R? Three. And then I said, it looks like I have B-A-R-R-I-E left. How many ways can I mix that up? Six letters, two R's. That was my first thinking, which was 1080 and I said none of these. But, what's 1080 divided by three? I worry that I'm double counting somewhere along the way. I think that's right, but somebody else presented an explanation to me that made me think there was another way of doing it. Then we do that way, I think. What did you do? What did you do? One option for the first? So, it has to be an R there, you said I have to put an R there. Okay, six, five, four, three, two, one, can't work because you got repeating letters. I'm going to argue that one. Okay, again, so once you put an R there, do two letters repeat, then I can't do the good old six, five, four, three, two. Because those repeating letters, I would have to go six factorial over two factorial. Yeah, but remember, this is like this, right? You didn't go, one, two, three, four, five, six, seven, eight, nine, ten, eleven. You didn't go eleven times eight times, you said it was eleven factorial divided by four I's, four factorial, four S's, four factor. You had to take into account that some letters repeated. I think once you pick an R, you have to take into account that there's still two R's remaining. I think it is going to be three and then six factorial over two R's, I think. Regardless, it's a bonus. Because I think that's a bit of a tough question. Which was, what did I say, one thousand eighty? I went with none of these. Sorry? What if you just circled none of these, then I guess you flew out and get the right bonus mark. I'll talk about how all the marking works out. In Square City, the city's blocks form the shape of a six by six square. All the roads of this city run either north, south, or east-west. Torrin starts at the northeast corner of the Square City and walks 12 blocks every day. If you walk a different route each day, about how long will it take them to have walked all possible routes? So this was a nice little twist on a pathway question. I said, it looks like it's going to be, if it's 12 blocks, if it's six by six. I said, I think it's going to be a word with one, two, three, four, five, six E's. And, excuse me, oh, south-west, sorry, not six E's, six W's, whatever. One, two, three, four, five, six W's and one, two, three, four, five, six S's. I said the total number of paths is 12 factorial over six factorial, six factorial. Oh, and Dina, because these two happen to add to 12, the shortest way to type that is actually 12 choose 6. That's just a nice coincidence. So I crunched that. I said, okay, what is 12? Come on, get that out of there. What is 12 choose 6? 924. So if he walks a different route each day, about how many years will it take him? I said, well, if I divide this by 365 days in one year, two and a half years. Nice little extension question. I thought that was kind of nearly cool. Matt over here says, how would you keep track of that? Good point. You'd probably do it systematically. In how many ways can four women and three men be seated on a bench? If the men and women must alternate seats. Oh, I have to start with the females then, don't I? It's going to be first woman, first man, second woman, second man, third woman, third man, fourth woman. One, two, three, four, five, six, seven. I have four choices for the first female, three, two, one. I have three choices for the second one, two, and one. It's actually four factorial times three factorial. Four, oh heck, four factorial is 24 times three factorial is six. It's 24 times six D, is that right? What is the seventh term of A plus B to the eighth? So they want a specific term. I could write them all out. By the way, what's my exponent? Eight, how many terms are there? Nine. I could write out and stop when I get to the seventh. You know what? I'm going to use the TK plus one equals N choose K, A to the N minus K, B to the K, and I'm going to do my list. A equals, oh A, that's quite convenient. B equals, oh B, that's also quite convenient. N equals eight. They want term seven. What does K have to be if we want term seven? That's the adjustment. It's going to be eight choose six, A to the two, B to the six. By the way, I would then cross out B right now. I would cross out D right now. I would cross out A right now. It's either that one or that one. Let's see. The coefficient is going to be eight choose six times that number there squared, but there is no number in front of the eighth and the visible one times whatever was in front of the B to the six, but there's only visible one there. So you know what? It looks like this time the coefficient is going to be eight choose six. Ah, C is the correct answer. Amrit, on your test, I'm not going to give you one without a coefficient. I'm going to put like a three or a negative two or something in front of one of these, okay? Any both? One, two, three, five, seven, and eight. We have to have odd digits. So if we're having odd digits, we can end in a one, a three, a five, or a seven. Why I wrote seven before the five? I don't know, but I did. Pardon me? Four digit numbers. It looks like I have four choices for that. And now we're allowed to repeat. Remember what that means, Madison, is you're throwing the tile back into the bag and starting to shake and scratch. No zero in here, so no restrictions. Of course, my seven really looks an awful lot. I go one, and it looks more like a seven. So it looks like it's going to be one, two, three, four, five, six, six. Is it six times six times six times four? Two hundred and sixteen times four, which is going to be eight hundred and sixty-four? Woo, math in my head, I'm back. What if repetition is not allowed? Okay. It's when repetition is not allowed that I find my scrabble bag, the one second it takes to draw that really helps, because I can make up an example. I can say, okay, we got four choices for the odd number, and I'll cross out one of them. Maybe it was a three, and then I can count. Oh, I got five choices left. Maybe it was a two. Oh, I got four choices left, three choices. I can really spot the pattern by crossing things out as long as I go. Is it five times four times three times four? So it's going to be twenty times twelve is going to be two hundred and forty? I like number two, I like number two, I like number two. Oh, what if number two was a multiple choice question? Would you actually bother solving it? I'm going to plug in your four answers in for N, but I'm going to give you one algebraic one like this. N minus two choose two equals sixty-six. Okay, I know from my formula sheet that N choose R is N factorial all over R factorial N minus R factorial. So this is going to be N minus two factorial all over two factorial N minus four factorial. Where did the N minus four factorial come from? N minus two minus two. Look up. Here is a common mistake that I see kids do. They forget to drop the sixty-six down, and so they end up with a quadratic, but because they've forgotten to write the sixty-six, they can't minus the sixty-six over, their quadratic won't factor because they've got sloppy. So remember, it's an equation. Drop the right hand side down two. Oh, and what is two factorial? Two. So on my next line, this is just going to be a two, not a two factorial. And I'm going to move it over to this side by multiplying by two. Dividing by two on this side is the same as multiplying by two on this side. I'm going to write this as N minus two factorial all over N minus four factorial equals 132. Move the two over times sixty-six. I can't cancel out the exclamation marks, the factorial, but I can simplify them away. Which one's bigger? Top or bottom? N minus two or N minus four? N minus two is larger. This is going to be N minus two times N minus three times N minus four. Oh, I'll stop there because I have an N minus four factorial on the bottom. I won't forget to write the 132 here and yay, it cancels. And I get N minus two times N minus three equals 132. I get N squared minus five N plus six equals 132. Minus 132 from both sides because this is a quadratic. How do I know? It's got a squared. I'll get N squared minus five N minus 126. I went numbers that multiply to negative 126 and add to negative five. I have no idea. But what I would do just to get a feel for what's out there is I would go 126 divided by two. Ah, what goes into 63? Nine and seven. You know what? What's 126 divided by 914? They thought you were skipping. They told me that you were playing soccer at lunch and they were going to say that you were bet. So if I'm factoring a big number, what I usually do is I try a nice small one and then I look at my remainder and I say, hey, what goes into there as well? So nine and 14 worked. You guys ever have deja vu? Nine and 14 will work and I think I'm going to have a minus 14 plus nine. This is going to be N minus 14 and plus nine equals zero. My roots are 14 and negative nine, but I can't have negative nine numbers. How would I give part marks out on that one? Well, if you got it right, you get full marks, but you got to do algebraically. I gave one mark if you were able to write the choose portion. I gave one mark if I saw the correct quadratic. One mark for the answer, but if you didn't reject the extraneous, I would take a half mark off. Pathway. Can I use my factorial shortcut here? No. I would have, if I could have turned my paper 45 degrees so that the A is right above the B. It's way easier, but here we go. I'm going to use Pascal's and I have to admit I find this one of the dirtiest, funnest lessons of the year. Here we go. One, one, one, one, two, one, one. What goes here? Three is coming in. Three is coming in. One, one, four. Skip, skip, skip, skip, skip, skip, skip. Four, one, one. I skipped a line. I missed the line with that four there. I put it in the wrong place. Note to self. Don't do that. The four goes right there, right? One, a five would go right there, Mr. Dewick. What would go right here? Four plus zero, a four, a five, and a one, a six, a five. What goes here? Nothing. Apparently you can't get there. If you felt better putting a zero there, go ahead, but nothing. Nine, six, eleven. What goes here? The five drops down because there's nothing right there. The nine drops down because there's nothing right there. And fifteen. Almost there. By the way, I find the most math mistakes occur on the last two lines, so raise your level of concentration up. Sixteen, fourteen, twenty-four, thirty, thirty-eight. Is the answer a sixty-eight? Let me check my answer key. Woo-hoo! There it is. Now, if you get sixty-eight, you get three out of three. Otherwise, how do I do part marks? I'll be completely honest with you. If you use Pascal's approach and you made one mistake, you get two out of three. If you use Pascal's approach and you made two mistakes, you still get two out of three. How do we get one out of three? You can't. How do we get zero out of three? Use factorials or some other non-Pascal's approach. In other words, you're almost guaranteed to get at least two out of three on this. Long as you can sort of show me that you're adding. Did I miss a question? I think I did. Seventeenth term of the twenty-third row of Pascal's triangle. Why that's twenty-two, choose sixteen. One less, one less. How do I know where the twenty-two goes and where the seventeen goes? I can't go from seventeen objects, choose twenty-three, or from sixteen objects, choose twenty-two objects, because you'll get an error. And the answer is twenty-two, choose sixteen. Seventy-four thousand, six hundred and thirteen. And I'm hopeful that the twenty-third row is far enough down that this year, no kids draw it out that far. I hope that question's been getting bigger every couple of years. So how about row nine? Some kid drew it. Go eleven. Some kid drew it. Row thirteen. Some kid drew it on the back of their test. So I keep upping the row every year. That little shortcut, not on your formula sheet, you can either memorize it. We did derive it by writing out Pascal's triangle, but I think that's one of the few times where I would say to you, I'll memorize the shortcut. It's one less, one less. At least three girls on the committee. At least three girls means three, or four, or five. Kelvin, this is a committee question, and it doesn't look like they're talking about the order making a difference. They haven't mentioned a president, or a secretary, or a treasurer. So I'm going to use bucket and choose. I have girls, ten, boys, eight. The first one is going to be three girls, two boys. Ten choose three, eight choose two. The second one is going to be four girls, one boy. The third one is going to be five girls, no boys. When you crunch the numbers, you'll get 5,292. How would I give up part marks? I would give out one mark if I saw this, and then one mark for the answer. It is 5292, is it not? Yes? And again, Nick, it's way easier to type in the first one, hit enter, and then go second function, write down the answer. Second function, enter, change the four, change the three to a four, change the two to a one. It's faster than just backspace. Is that a hand up, Elizabeth? No, that's just hair number 93 is out of place. Okay, find and simplify the term containing x to the 14th in math. So if there's talking about a specific term, I'm definitely going to be using tk plus one equals and choose k, a to the m minus k, b to the k. The problem is I don't know when this term, I don't know what term number this is. First one, second one, third one, tenth one, 12th one, because there's 12 terms. I don't know. Well, this is a, I mean, this is b. We're looking for someone? Okay, need me for a second? Okay, hold on, folks. I'm going to pause the video. Right click, pause. So the problem here is, I don't know when this term appears. What I'm going to do is I'm going to try and figure out the pattern. So I know with a 12, I'll start with an 11 right there. I know it's going to be a to the 11th, a to the 10th, b, a to the 9th, b squared. And I said to you, you'll all be able to figure out the pattern from three terms only. We could do more, but I'm not going to. And all I'm interested in right now is the x's. How many x's would there be in the first term? Zero. How many x's will there be in the second term? Well, I'll have an x squared to the one. You know how many x's I'll have? Two. How many x's in the third term? I'll have an x squared, squared. You know how many x's I'll have? It looks like I'm going up by, here we go, close my hands, zero in the first term, two in the second term, then the third term, then six, then eight, then 10, then 12, then what term number will have an x to the 14th in it? The eighth term. Ha-ha! I want term eight. Really, that's the hard part. After this, it's now going to be the plug and chug into the t, k plus one equation, where a is two, b is x squared, n is 11, and k is not eight, what's k? Seven. It's going to be 11, choose seven, two to the 11, take away seven, fourth, x squared to the seventh, and I smile because I can see the x to the 14th is going to pop out. I've done this right. Let's get the final, simplified answer. It's going to be 11, choose seven, times, oh, and there's going to be a two to the fourth. I'm going to get 5,280, x to the 14th. There's your final answer. How did I give up part marks? I gave one mark if I could see this somewhere, and one mark for the answer. Yes. You put x squared as a? Okay. So Nick says, Mr. Duk, here's what you're really asking. It doesn't matter if I change the order. a plus b to the nth is the same as b plus a to the nth. However, and you probably noticed that already in grade nine when you were doing lots of foiling, you may have noticed, oh yeah, the x at the front or the end, you're going to get the same answer, but I don't know if I would use that approach, but yeah. No, you can have, yeah, we can write it like this. Now, you're also recognizing we normally don't write binomials like this. Usually we go variable and then constant, but you know what, it's a nice universal flexible equation. It works. If you haven't figured it out by now also, Nick, I love good math shortcuts. Even this one is, although it's complicated, it is a great shortcut. Number seven, I told you this constant term would rear its ugly head. Number six, except the constant term means how many x's do I want when I count on my fingers? Zero. I need to figure out when that's going to occur. That's going to be a, that's going to be b. Because there's no exponents on the x, my gut is it's going to occur on the fourth term or the fifth term, halfway along towards eight, but I'm not quite sure. Let's see. I would have a to the eight, a to the seventh, b, a to the sixth, b squared. I would have x to the eighth in the first term, x to the seventh, one over x to the one in the second term. How many x's is that grand total? Six. I think I'm going down by twos. Just a guess. I would have x to the sixth, one over x squared, squared, squared, six on top, four on the bottom. Sorry, not x squared mister do it. There's no squared there. Six on top, two on the bottom, four. Yeah, you know what? Eight, six, four. I'm going down by twos. Start with eight. Then we have six. Then we have four. Then we have two. What term will have none? Term number five. Aha! I want term five. And David, right above this, I'll write the dk plus one equals n choose k, a to the n minus k, b to the k. And I'll start listing my data. Here we go. A is x, b is two over x, n is eight. K is, careful, four. Eight, choose four. x to the fourth. Two over x to the fourth. Yeah, I noticed my x's are going to cancel. My coefficient is going to be eight choose four times two to the fourth. Eight, choose four times two to the fourth. Eleven, twenty. Yes? Nothing? Yeah? Oh. I thought I heard. Here's how I gave out the marks. If you figured out that it was term five, I gave you a half mark just for that. If I saw that, you got a half mark. I guess for some reason I had the kids writing last year 70 times 16. This year we're doing it on our calculators, so anyways, full marks if you got the eleven twenty. Number eight, right at the first four terms of that expansion, half mark for each term, and unfortunately because it's a half mark for each term, they're all or nothing. I'll read them out to you. You should end up with a 512 x to the ninth. Minus 6,912 x to the eighth y. Plus 41,472 x to the seventh y squared. Minus 145,152 x to the sixth y cube. So if you missed a term, each term you missed or got wrong is a half mark off. Nick, what do you want me to read to you? You're squinting? Want me to read it again or first term, second term? I know it's my writing too and I'll, you know, it's messy. Nick, what was the exponent nine? So it's going to be 8 to the ninth, 8 to the eighth, b, 8 to the seventh, b squared, 8 to the sixth, b cubed, and then 9 to the 0, 9 to the 1, 9 to the 2, 9 to the 3. And then be careful with your calculator. Lots of typing and stuff. That's right, yes? And then I think I changed the name on your quiz. On your quiz is it Marissa? Yeah, I changed it every couple of years for a different student, but I forgot to change it this year. Otherwise I would have put one of your names there. Sorry. Anyways, the original when I typed this up was Allie. Holy smokes, that was a while ago. So she is a, Marissa is a disc jockey. She has chosen to play ten songs for the school dance. Marissa creates one ten song compact disc for these songs. How many different ways can she arrange Kelvin? The word arrange permutation. Ten factorial. Which is also ten p ten which is three, six, 28, 800. How do I get two marks for that? I guess that one's kind of an all or nothing. Marissa realizes that the first three songs of the CD are important in setting the tone for the evening. Oh, of course they are. She chooses three problems from the original ten. Oh, look at this. Math is wonderful. The never ending algebra problem. In you plus me equals us. Oh, look at that. How could that evening not be a phenomenal dance evening? How many ways can those three songs be placed at the start of the CD? So just those three songs I think you could go like this. Three choices, two choices. One choice or three factorial. I think there's six different ways just to mix up those first three songs. So how many ways can she burn the CDs? Yes, this was made back before mp3 players. Sorry. How many ways can she burn the compact discs if these three songs must be at the start? So it seems to me we have the three songs. How many ways were there to mix those up? Six times. How many songs are left? I think it's going to be six times seven factorial because you can mix the other ones up whichever way you want. It's going to be silly. You know what? There are 30,240 different ways to start that evening off the way it should start off. With some math music. Oh. 30,240 ways to increase your mathematical enjoyment. How are we marking this? Now, I had said that multiple choice was 1.5 each. So there is a total of eight multiple choice. What's eight times 1.5? I know. What's eight times 1.5? 12. Minus the bonus 1.5. So this is actually out of 10.5. And then we have one, two, five and three. This page is out of eight. One, two, three, four, five, six, seven. This page is out of seven. Two, four, six, eight. This page is out of eight. So I have eight plus seven is 15. Plus eight is 23. Plus 10.5 is 33.5. I gave everybody a free half bonus mark. And that's why this quiz is out of 33. Is that okay? So give yourself a score out of 33. Do I see you between now and the test? So I don't actually want the quizzes. I do want your quiz scores. So the person at the end of each row do the usual. Pass them in please. So questions from the review that I like. Number one, I like. I like number four. I like number six. In fact, why don't I get really clever, Mr. Dewick? You have a one that you can write on right here. So what did I say? I like number one, I think I said, right? I like number four. I like number six. Seven, fundamental counting principle. So I'm going to give you something like number nine where you have to simplify a factorial. Number nine is a tough one. It's on the upper limit of how tough it'll be. It may be easier than that, but if you're not sure what to do for number nine, look at my online answer key. Something like number 11, there's going to be some kind of a digits question. How would you do number 12? I think it's asking how many different ways can you mix up four C's, three I's, and three O's because there's ten people. This would be ten factorial over four factorial, three factorial, and you can turn it to a word. I'll circle number 12 because it's definitely something like number 13. Either an NP2 or an NC2, there's going to be some kind of a choose equation to solve algebraically. This one here is multiple choice. I trust that you would just plug in those four numbers and see which one worked out to 42. I'm going to give you one on your written. I like 15. 16, pathway questions. 17, 18. 19 is a regular pathway. You can use factorial shortcuts. I said 17, 18, 19, right? You can use factorial shortcuts. 20. I think I've already given you a couple of these. Find the whatever term of whatever. You don't have to do all of these, but I'm saying make sure you know how to do these ones. 22. 25 is very similar to 20. You need to be able to find a term. Something like 26, where I'm going to actually say you need to know not just how to find this on your calculator. You actually need to know what it looks like in the notation from the formula on your formula sheet. Okay. Number 27 is finding a term. I've already given you about three of those already. Three or four of those, I think. So I'm not going to circle number 27, but you better know how to find a term. 29. Kelvin, if you look at number 29, that's the permutation, not a choose. So 7p3, not 7c3. Pathway. I'll do that. I've given a couple, but I'll give a couple more. By the third term, I've already done a couple of term number ones. Something like 33. Here's a simplifying factorial one, 34. 35 going backwards. I'm going to give you one. We're asking you to go backwards. 38, a digit question. Well, I'm not going to circle 39. Look at number 39. What's the answer? 10. Okay. So I'm not going to circle that, but I like that question. I am going to circle 37. More factorial simplifying. So if you need more practice, there's going to be a factorial simplification. Definitely an irregular pathway problem question. We have to use Pascal's. Sorry, I might just throw this story. So something like number 43, except it's going to be a written question. It will be the first four terms. There's going to be an at least or at most question. 49, Kelvin, permutation. Does that help answer your question? I hope. Could you find the eighth term of that using the take tk plus one? I've already given you like four, so I'm not going to circle that, but make sure you know how to do a term equation. I've done a few irregular pathways already, so I'm not going to circle that one. What's the answer? Number 55? Eight, right? I already gave you one like 56, I think. 58, 59 is going to be some kind of a committee question. So written. I may also, instead of giving you the factorial equation in choose notation, I may give it to you already in factorial notation. By the way, that three factorial, what would I make it on my next line? A six, and then I would multiply by the six to move it over and make it a 30. I don't like number four, by the way. It's kind of, it's no whatever. 11, number five. So if you're having trouble with factorial equations, because I've told you that I'm going to assign them six and seven are fair game. I'm not going to circle them, but if you had trouble doing number four, or number three, sorry, do six and seven. I showed all my steps in my work. Nuke number eight, number nine, number 11. By the way, 10 is the same, like it's also two groups picking at least. So it's very similar to number nine. 12, three groups, colors that very similar to number nine. 14, actors, males and females very similar to nine. So there's going to be some kind of a question like that. I won't circle them, but if you're having trouble with those, there's more practice for you, and I'll give you one more permutation question to solve. Okay. And again, just a reminder, my solution key does show my steps pretty good, I think. You should be able to figure out kind of what I was doing, and this is online under under block H combinatorics, and it should be right near the beginning of all the files. Almost. Any questions? So there's about 20 minutes left. I'm purely going to pause.