 Number one how many odd three-digit numbers are there well once again I didn't specifically say whether it involves repetition or not Usually if they don't say you assume you are allowed to repeat. How come we're late, sir? Thank you If you look at number one, it doesn't specifically say whether without repetition Probably if they don't say you can't repeat you probably are allowed to because it's not clear here again, Blaine I'm gonna take both answers So if I'm allowed to repeat an odd number has to end in a one a three a five a Seven or a nine and I'll visualize my little Scrabble bag five six seven eight So there's all the numbers in the Scrabble bag How many choices do I have here five and because I'm saying you can repeat We'll put the number back into the bag give the bag a shake. How many numbers do I have here? Nine why not ten? I can't start with zero put the numbers back in give it a shake. How many choices do I have here? Ten so if you did 450 I would accept that now if we're not Repeating what would it be? Well, you would still have five choices because it's got to end in an odd number and Let's suppose you picked a five for the very very end. How many choices do I have for the first number now? Eight not nine because I can't start with a zero. So let's suppose I put a one right there How many choices do I have now? eight 40 times eight 320 I Would take either of those answers because the question isn't clear. Oh, by the way if there's that many odd numbers grand total and if there's that many that don't repeat how many repeat Think a hundred and thirty must have like two of the same number in them because this was all of them Including the repeats. This was not including the repeats. Whatever the difference is left over How many ways are there to arrange the letters of the name do it five factorial hundred and twenty? How many ways are there to arrange four letters of the name do it? You could go four blanks five times four times three times two times one or it's also five permutate four and I think you get a hundred and twenty again, don't you oh Vandervelden okay now we're giving you a challenge one two three four five six seven eight nine ten eleven twelve twelve factorial all over two V's to ends Two D's Three ease. I'm gonna go twelve Factorial and I'm just gonna go to divided by eight times six Is that right two times two times two is eight three factorial six faster typing and you get Wow nine million nine hundred and seventy nine thousand two hundred and this brings us to our typo What I should have done is I should have made instead of having it be choose three It really should have been choose two if you have a choose three here. I first of all said, okay I'll put it into the permutate equation So it's going to be the top thing divided by the top minus a second thing factorial What is n minus one minus three if I tidy that up? And minus four so I got n minus one factorial over n minus four factorial Common sloppy mistake by the way Chelsea kids forget to drop the two hundred and ten down and Then when they end up here with the equation they forget there's a number there And so they're quadratic won't factor because they've forgotten to subtract the number over trust me I've seen that happen a gazillion times as well. So just don't be sloppy. It's an equation. It's not a simple It's an equation now when you do this I said which one's bigger n minus one or n minus four n minus one is larger than n minus four So I started to expand it and I got n minus four factorial and I actually ended up with an x cubed or an n cubed equation. I Failed it out very very quickly Minus the two hundred and ten over and that's why this is a typo you can solve this using math 11 remember doing the root Rational root theorem the remainder theorem the factor theorem and polynomial division or synthetic division all that or you can use the Polysmoke program that I've given some of you on your graphing calculator that works fine, too But a x cubed equation like this is not considered fair game for math 12. So here's the deal if you ended up with this equation and You got eight as an answer you can give yourself two bonus marks If you just guessed your way to eight I'll give you one bonus mark, I guess Okay, that number right there that three should have been a two if it was a two you get a lovely quadratic There is going to be some kind of a factorial equation on your test and your test is going to be a week from yesterday No, you're block D. Your test is going to be a week from today Tutorial Monday after school and I'll send an email out tonight as well Okay number six To simplify number six. I said I'm pretty sure these two go together and these two go together So in my mind I drew a line kind of separating them and I said 102 factorial is bigger than 100 This is 102 times 101 times a hundred factorial the hundred factorials cancel Then I said which one's bigger n minus four n minus two n minus two is larger than n minus four So I started to expand the bottom and I got an n minus four cancel cancel So I ended up with 102 times 101 divided by n minus two times n minus three I did multiply the 102 times 101 together to get 10,000 302 over that I gave one mark for the top one mark for the bottom So there is going to be a simplify some type of weird factorial expression It'll be multiple choice on your test though. I Like number seven like number well, honestly, I like every question on here except for the typo in number five But I still like the idea of number five, but with a two instead of a three You're gonna see almost every one of these in the future in some variation So committee question The first one says how many ways can you select a president a secretary and a treasurer this is not a bucket question because they're not talking about different groups and It looks like the order matters because if you pick first you're the president if you pick second You're the secretary you pick third you're the treasurer So I said this is 11 p3 or you could have gone three blanks 11 times 10 times nine This time I felt comfortable using the shortcut. I was pretty sure I was right 990 Any three people for the committee so all you want to know is whether someone's on or off the committee You don't care about what position they're in or order doesn't matter. That's 11 choose three Which was 165 and then if they break it up into groups and order doesn't matter That's when I go to this bucket and I draw this every time because it takes all of one second So I said girls and boys four girls seven boys. They want me to choose two I'm gonna choose one and the equation just pops right out 42 and the last one at least What is at least one girl mean? What are the cases that include at least one girl one or two or three? So I made a little heading I got organized. I used the same bucket from up here I didn't redraw it although on a test. I probably redraw it just so I didn't make a sloppy mistake, but I got 130 oh And remember Kellen to type this I said don't type this whole thing in do the first one Hit enter and then go second function enter and just change that number and that number to get your next term It's way faster Although I'll be honest I actually skipped this one because I just done this one on the previous line that will often be the case on these Questions often you'll have done one of them on the previous line. I Gave one mark if I saw this line here and then one mark for the answer Okay Turn the page Order matters in a permutation Does not matter in a combination Can I abbreviate those as perm and comb I guess I? Probably wouldn't on a test though license plates pause while I Take care of my late student license plates If repetition is allowed it's four letters and you're putting the letters back into the scrabble bag So it's going to be 26 26 26 26 I said that's 26 to the fourth faster typing and it's 10 10 10. I said that's 10 to the third and I got 456 million 976,000 and I believe Toronto on like Ontario. I believe has gone to four and three BC looked at that and instead they decided to allow letters and numbers to interchange Well, Ontario run out of license plates anytime soon if they can have 456 million nine hundred and six thousand Be fine Even if you remove the ones that you can't have because they're either bad words or I'm guessing nobody has a 0-0-0 or maybe the Premier does right? What if repetition is not allowed now repetition is not allowed That's actually the shortcut for that was a permutation You could have drawn four blanks and three blanks and gone 10 9 8 and 26 25 24 23 but a permutation was what we were doing when we were not allowing repetition Either of those will give you 258,000 330 sorry 258 million 336,000 And the last one was a pathway. It was a regular pathway question. I got 56 Oh, and remember there is a shortcut since three and five add to eight You could have just gone eight choose three That doesn't work for 3d But it works for regular pathway problems Can you give yourself a score not out of 18 but out of 16? But you might get the two mark or the one mark bonus add them up Pass them forwards, please pass them inwards, please. I mean We're gonna finish off the unit today. Can you open your workbooks, please to page? 400 and 12 412 Yeah I'm gonna finish off the whole lesson because that might answer the question and then we'll tackle it from there Okay page 412 and then shush So last day we looked at the very impressive sounding Fundamental no not fundamental counting principle. We looked at the binomial expansion theorem It was a great shortcut and first we learned the pattern if you just wanted to foil something out So we said if you were foiling that out that six there told you eight of the six eight of the fifth eight of the fourth Eight of the third a squared and oh that B B the zero B to the one B to the two all the way up to B to the six And then that six there said the coefficients were gonna be six two zero six choose one six choose two etc And then we also did say and the reason I said turn to this page here as we said You can also find a single term the last couple of questions that we did we said what if you just want to find one? Turn you don't want to find all of them Well, you can use this and we said the formula looked kind of ugly and the reason was we had to do an adjustment This k plus one here The reason we had to have a k plus one is because we're starting to count from zero in our chooses Somewhere we were gonna have to do some kind of an adjustment and it was easier to do an adjustment here Then it was to do it here And so this is a formula that I would put a little box or highlight around it is on your formula sheet I am gonna ask you to use it on the test and we're gonna practice one or two of these today and Then we're gonna go backwards and we'll have finished off the unit Okay, so can you turn to page 413 and we're going to do B To be together Now to be says find the middle term of that a Lot of people when they see this go. Oh mil terms three hang on Kellen. What's my exponent? Six now do you remember from last day if your exponent is six how many terms are there not six seven So if you have seven things in a row What's the middle term? It's the fourth one because you got three in the front and three in the back So here's what I said to you last day because they're asking us to find a single term I always write down this tk plus one equals and choose k a to the n minus k B to the k it takes one second I copy it right from my formula sheet But I learned over the years folks taking the one second to write that down gets rid of most of your sloppy mistakes And then over off in the margin my physics twelves will will grant a defect. I'm gonna list my data So what's sitting where the a is? to a What's sitting where the B is don't say five? negative five What's and this is the easiest one that kids always forget six and Then what term number did we want to find term number? What? So I always write down term for right here, and then I can say if I want term for what's k It's not for what is it really ah? K is gonna be three And now it's Complicated plug-and-chug, but it is plug-and-chug It's going to be and six Choose k three to a To the n minus k in your head, please three negative five To the three and then I'm going to go to my calculator and do all of the numbers So I'm going to go term by term here certainly in the front. I'm going to have a six How do you get what what term number do I want? term number four What does k have to be to get a term for there? Okay, that's that's why Justin that's why I always write this above if I do this and I write that 99% of the time I'll catch. Oh, yeah, okay. It's not for three. Okay That all right. Okay. Okay. Okay. Okay. Are you k okay? Okay? Numbers, so I'm going to go look six choose three Times now in here. Can you see I have a two cubed? Two to the third times and here I have a negative five cute Negative five to the third The coefficient is going to be Apparently negative 20,000 Did I get all the numbers? I think I did yes negative 20,000 What are my variables? eight of the third and The second term didn't have a variable in it And again, Leslie, I'm going to say to you I defy you to find a better shortcut to do this in one line Otherwise you would have had to write out all six Multiply the last two get an answer multiply the third one in get an answer Well, the fourth one in get it all the way up till until you had all six of them And then gather like terms and hope that you hadn't made a math mistake by then and then you'd find the center one Is that okay? jogs and memory Okay, I like number four. I like number four. I like number four Number four is what we've just done but in reverse Number four, here's what they have said One term of X plus something to the 10th worked out to that What does that something have to be? Now the only thing I don't like about this is what letter did they put right there? a What letter did we put right there in our binomial expansion theorem? Let's all of us right now make that a b and that a b so we're not confused silly Sillily a word Because they're asking us to find one term. I'm gonna go term k plus one equals and choose k a to the n minus k b to the k You have to memorize that Sally. No, it's on your sheet next to ncr and PR Make sure you know where it is though And I'm gonna list What's sitting where the a is? x What's sitting where the b is? b What's m? 10 Here's my problem. I don't know what term number this is. I don't know what term number this is except You see that x? That's sitting where the a is don't write this down. Just watch. I'm gonna make a little argument here. I Think the first term would have had an x to the 10th in it because there's a 10 there And it would be a to the 10th the next term would be A to the 9th except a is x it would have an x to the 9th in it for the next term having it X to the 8th, right? Okay. I Think I can figure out What term this is all of you get your 10 fingers out get to count on your fingers today. Are you ready? You need 10 so the first term has an x to the 10th. Do you agree with me because it would be a to the 10th What would the second term have x to the then? then then Then when does the x to the fourth appear term number? What ah, huh? See how I got that we counted our way there and Justin That's going to let me write this magic thing right here term seven because soon as I write term seven. What's K? Six I don't know of another easy way to get there aside from counting on fingers. What if the exponents 20 well? Use your toes I guess but Figure it you'll be able to count Okay What's n 10 choose? What's K? six x to the Fourth hey, which is what I wanted see it. Yeah, okay good good B to the 6th. What is 10 to 6? 210 okay, so I'm going to write this 210 x to the fourth B to the 6th But unfortunately right now that is not enough If I write don't write this next part down if I put term seven right there I got a problem wait wait wait wait wait wait wait. I'm not going to put term seven right there I know what term seven worked out to they told me what a term seven worked out to What I'm now going to do is I'm gonna say this was three million two hundred and eighty one thousand 250 x to the fourth See, I'm working my way backwards very very carefully. Why is that so nice? What happens to my x to the fourths? I'll take that and How would I get the B to the sixth by itself? How would I move this two hundred and ten over? divide by two hundred and ten divide by two hundred and ten and I'm gonna get What is that thing divided by two hundred and ten is it fifteen six twenty five? Okay, I get fifteen six twenty five equals B to the sixth. How do I get rid of a sixth power? Sixth root which is on my calculator We're oh in the math submenu. So the final answer B is gonna be the sixth root of fifteen six twenty five and With an ending like five, it's gonna be five or twenty five. I'm almost certain maybe 15 But I'm pretty sure it's gonna be five or twenty five. How do I do six through six math? Oh There's the end through option five of 15 625 You know what my original binomial was x plus five to the tenth That's what would give me Somewhere along the way a three million two hundred eighty one thousand two hundred fifty x to the fourth So I am gonna give you one question and it's gonna be a written question Where I ask you to go backwards where I say here's what the term worked out to you You'll have to look first of all at the exponents and count your way to figure out what term number it is and Then carefully substitute everything into the term equation until eventually you'll end up with Something to an exponent equal something how to get rid of the exponent take the root. I like that question. I Like that question. I like this question. I like this question example five. I like And then we're done back then we're done the whole unit Number five has some vocabulary it says this Find the constant term i.e. the term independent of x and the expansion of that What the heck do they mean by this do you notice this first term has the x's on top Do you notice the second term has two x's on the bottom? Somewhere along oh by the way, what's the overall? What's the exponent? 15 how many terms are there? 16 somewhere in those 16 terms because you have some x's on top and some x's on the bottom Somewhere all the x's will cancel And when all the x's cancel you know what we call that a A constant term or we say Connor it's independent of x So what this question is asking is? What term is it but first I need to know when it occurs You see why x is on top x is on the bottom somewhere. They are going to cancel You say this again x is on top x is on the bottom somewhere. They're going to cancel That's going to be a that's going to be b and what I'm going to try and do is figure out when this term appears Do the x's cancel on the first term on the second term on the 10th term on the 15th term when do they cancel? And to do that kellen i'm going to look for a pattern What's the exponent? So i'm going to write a to the 15th a to the 14th b to the 1 a to the 13th b squared because if I can't spot the pattern after three terms I probably don't deserve to be in that 12 But i'm going to go even more Simplified than that All I care about right now kellen is finding out when the x's cancel So even though there's a two here. I'm not going to worry about it temporarily And even though there's a negative one on the top. I'm not going to worry about it temporarily All I want to know is when do the x's vanish? So I'm going to argue that this first term would have An x to the 15th in it. Yes. I know it was a two cares Because a to the 15th and there's a there's a one on this x This second term would have an x to the 14th times It would have a single x squared on the bottom Because b has an x squared on the bottom Stephanie how many x's on top grand total how many on the bottom how many left and where I wonder Steph if the pattern is that these are just going to go down by three if that's the case I can count my way there and figure this out in other words. I wonder if this is going to be an x to the 9th Because it looks like it's going down by three. So let's find out. Let's find out. Okay So here I would have an x to the 13th Times and I would have a one over x squared squared How many x's on top martin How many on the bottom when you square the square how many left and where? Have we spotted the pattern after three examples it looks like we're going down by threes Get your fingers out So the first term Hannah is going to have x to the 12th second term is going to have x to the 9th then Sorry 15th x to the 15th. I'm sorry x to the 15th then x to the 12th then nine then six then three Then when do I have zero x's when do the x's cancel what term number? aha Okay All that work was to tell myself. I'm finding Term six. Oh, but above it. I'm going to write term k plus one equals n choose k A to the n minus k b to the k I guess you can go michel I'll do my little listing my data Going what's a Two x and now we're going to do it properly I ignored the coefficients because those weren't going to change the exponential pattern And that's all I was interested in now. I'm going to do the coefficients. I agree What's b careful? It's not one over x squared Negative one over x squared n is 15 k is Justin can you tell me what k is? Five By the way, I guarantee you if this is multiple choice I will have one answer for k equals six because that's the most common mistake And now it's plug and chug complicated plug and chug the plug and chug It's going to be 15 choose five Two x to the 10 negative one x squared To the fifth By the way, sally, how many x's will I have on top right here? Sorry, I'll scroll up so you can see because Justin's in your way. How many x's will I have right here? How many x's will I have on the bottom right here? How many left we we got the right term? It's the constant term. It's the term independent of x all I need to do now is the numbers So the number is there's going to be a 15 choose five times Two to the what what's the exponent on that two sitting inside the brackets? two to the tenth times negative one to the what what's the exponent sitting on the negative one Five now. What is negative one to the fifth power? Is it negative one or positive one? So I'm going to be a bit lazy chelsea I'm just going to go time negative one without adding the exponent because I can do some math in my head negative three hundred and seven no negative three million seventy five thousand seventy two 307 5072 So I'm going to give you one like this this one probably multiple choice The previous one though definitely as a written question and That I think I said to you guys in the last lesson This shortcut does have a few curve balls. These would be the two kind of curve bally questions. You need to practice these so I already assigned number one And I assigned seven a and seven b And c and e is that right? Yeah, and eight so I think now I can also say try nine b Now they've told you that you want to find the term that has the x to the fourth y cubed in it You'll have to use the counting on your fingers trick to figure out when that term appears when well What would the first x have on an x to the what? What would the first x have on an x to the what? seven then Then okay, you can figure out when the x to the fourth will appear count on your fingers That'll tell you the term number and then use t k plus y equals n choose k a to the n minus k b to the k 10 Feel free to change that a to a b if you want to because I just don't like the fact that they put an a where I'm used Having a b. Why can't he has a 11 and I think I already signed 12. Yes, I do So there is the homework part one and I have for you And unfortunately you're going to have some homework this weekend folks But the goal is for your five-day weekend to have no homework for the Easter weekend I have for you a big take home quiz summary of the entire unit as well And that's what we're going to be going over first thing on Monday. Okay, so let me hand that out