 We'll come back to our lecture series math 1050 college algebra for students at Southern Utah University As usual, I'll be your professor today. Dr. Andrew Mistledine. This brings us to the final section We have about polynomials in our series at the end of Chapter four here and what we've been spending a lot of time talking about factoring polynomials But the last thing I want to talk about in this in this unit here on polynomial functions is Multiplying them because factoring is a challenging process but multiplying can be a little bit difficult as well And so in preparation to the so-called binomial theorem, I want to introduce to you the binomial coefficients So what are these? So first let me remind you if you don't are just tell you if you don't know Let's talk about the factorial function So in factorial if you have a whole number a positive whole number in factorial is defined to be in times In minus one times in minus two times in minus three all the way down to three two one So you stick the product of all the numbers from one up to n So for example seven factorial is equal to one times two times three times four times five times six times seven That's that's this number right here. This is this seven factorial in factorial Now it's the product of one up until the number if it's a positive number if the number is actually zero zero Factor we define to be one And we'll make some more sense of why that is in a little bit People sometimes think that zero factor should be zero because shouldn't you be times it by zero? But you can't multiply from one up to zero because zero is actually less than it Zero factor we define to be one and the basic idea behind is what is a factorial? You think of it is how many ways can I line up children to go to the drinking fountain in kindergarten? When you're in kindergarten being the line leader is a big deal So whose first seat matters so if you have for example like five kids in this very small class That have to have to line up well basically you have five options Who's gonna be the line leader then the next slot there's gonna be four options right there because well? You know who's in next in line is matters because I mean they'll be the line leader tomorrow They'll be the first one to get the drink tomorrow. So status is a big deal here So you have five options for the first kid But the second kid there's four options because you don't pick whoever the line leader is Do you have three options for the next one two options for the next one and then one option for the next one Right here. So this right here is five factorial. There are five factorial ways, which is 120 There's five factorial ways of lining up five kids to go get a drink of water saying kindergarten now Coming to back to zero factorial. What if you have no kids in the law? There's no kids that came Maybe because they all stayed home for the coronavirus or something like that I don't know but if you have no kids then how many options do you have well? It turns out you might think like there's no options But when someone says there's no option what they really mean is there's one option no choice When someone says no choice that actually means there's just one option Choice would mean that you have a diversity of options and you can pick between them. So you have one option So lining up zero kids. There's only one option there and so that means zero factorial is gonna be one So the little of explanation there, okay So what do factorials have to do with our binomial theorem? Well, we're now going to introduce what we call the binomial coefficient So this is denoted as in over r. It's kind of like a fraction, but there's no fraction bar It's gonna be blank right there in over r, but we do put parentheses around it. That's mandatory This symbol right here is called the binomial coefficient and it's defined to be in factorial divided by our factorial and in minus our Factorial so why is that? Well, let's do a few examples of this So when you see this binomial coefficient you read this as three choose Three choose one because this is about choosing things. I'll talk about more than just a second So three choose one is gonna be three factorial over one factorial times two factorial notice that two plus one equals three So three factorial looks like three times two times one one factorial is just a one and two factorial is two times one So you always get this nice Cancellation here. You're gonna get two cancels one cancels. So you left with you're gonna be left with three over one Which is just a three So three choose one turns out to be the number three. I'll explain again some meaning to this in just a second If you did like in choose one factorial, what that would look like is in factorial over one factorial times n minus one factorial well in factorial looks like n times n minus one times n minus two All the way down to three times two times one one factorial is just a once a divide by one really doesn't do anything But then in one in minus one factor will be in minus one times in minus two All the way down to three two and one and so you can see there's a lot of cancellation here The one cancels the two cancels the three cancels the in minus two cancels the in minus one cancels Basically everything cancels out except for the in on top and the one on bottom What's division by one again doesn't do anything you need you're gonna get in right here and So this is a nice little thing you can simplify in choose one is always going to be in If we did for example for choose two this one's a little bit interesting calculation here For choose two looks like four factorial over two factorial and two factorial two plus two is equal to four The sum on the bottom should add up to be the number on top So four factorial it looks like four times three times two times one two factorial is two times two times one And we get it twice So that will cancel the two factorials and then the one divided by one doesn't ever do anything now two does go into four Two times and so in the end this gives us a two times three, which is equal to six Something that's interesting about these binomial coefficients is that even though there's division This thing will always turn out to be a whole number no matter what it'll always be a whole number look at six choose three For example, you get six factorial over Three factorial and three factorial one trick I want to show you is the following six factorial is the same thing as six times five times times four times three Factorial three factorial of course is three times two times one And then you have a three factorial bottom the notice that they're always gonna have a bigger factorial on top versus a smaller Factorial and so you could always factor the number on top using a factorial and so those factorials will cancel out So picking the bigger of the two factorials if there was one you can cancel that out So now we have a six times five times four over three times two times one now We have to look for numbers will notice three times two is six. So if you take three times two That'll cancel the six on top and we're left with five times four, which is going to equal 20 And so this is evidence here that this these binomial coefficients always turn out to be whole numbers You can always cancel all devices on the bottom here now if you take in in choose in right This is gonna look like in factorial over in factorial times zero factorial in which case the in factorials cancel out You're left with zero factorial Which we don't want this to be divisible. We don't want to divide by zero, but zero factor is not zero It's one so this is one over one which is equal to one On the other hand if you look at in choose zero right here This is going to turn out to be in factorial over Zero factorial times in factorial notice here that in fact or still cancels you get one over one One over zero factorial again, which is just one. So there's these numbers turned out to be the same thing So let me give you some explanation about these these binomial coefficients Now one thing that we just observed is the following that if you have the if you have for example In choose R. This is the same thing as in factorial over our factorial times in minus our factorial Which is the same thing as in factorial over in minus our factorial times our factorial Which is the same thing as and choose in minus R So if you swap the number so notice that R and in minus R add up to be R If you switch the number on the bottom to its complement there, you always get the same thing This is why in choose in and in choose zero turned out to be the same number because zero plus n equals in if I were to do something like Seven choose two. I know that this is going to equal seven choose five because five and two both add up to be seven We know those things right there. So that's a pretty nice observation Another thing I want to mention about these binomial coefficients here is that the binomial coefficient in choose R This is equal to the number the number of ways of Choosing of choosing our objects from a collection of n objects So for example, we go back and look at these numbers when I look at something like three choose one If I have three options in front of me if I have strawberry ice cream chocolate ice cream and vanilla ice cream I have to choose one flavor to eat. I have three options I could choose strawberry chocolate or vanilla, right? Those are the options. So three choose one is three in choose one If I have in options and I have to choose one of them Then there are in ways of doing that you just pick the option you want now If you have if you have more choices to make it gets a little bit more complicated if you have four objects And you have to choose two of them. There are six ways you can choose those two objects Or if there's six options and you have to choose three There are 20 ways of choosing three things from six If you take up if you have in choose in many objects, you have in objects You have to choose all in of them. There's only one way to do that. It's to choose everything If you have in object and you have to choose none of them There's only one way to do that. It's to choose none of them. So you have one option This next number right here 52 choose five like if you're playing a card game And you have to deal out hands of five five cards from a standard deck of 52 There's going to be 52 choose five ways you can deal out a hand and that's going to be 2,598,960 So if you're a card playing individual And you like to play by the odds This is a number critical to help you determine the odds of getting certain hands and things like that On the other hand, if you have five choose 52, what is that going to be? How in the world can you choose 52 objects from a set of five? That's not possible So this number turns out to be zero in that case If we write again that formula I had on the screen in choose r versus In choose in minus r Why are these two numbers the same thing because if you have in objects and you have to choose r of them By choosing r of them there were in minus r that you didn't choose And so choosing though like so let's say you're playing like a dodge ball game Right, you have to choose which of the in kids Which r of them are we going to choose to be on our dodge ball team? Well by choosing the r people to be on your team Essentially, that's the same thing as choosing the in minus r kids to not be on your team And if you've ever been that kid who didn't get selected to play dodge ball Then you know exactly that by not being chosen. You're actually chosen, right? You're chosen for something different. Sorry for that that very sad metaphor right there And so if we asked want to ask ourselves the question how many different committees of three people can be formed from a pool of seven people So we have to choose three people from seven That's going to be the number seven choose three And so by construction that's going to be seven factorial over three factorial divided by four factorial So seven factorial seven times six times five times four factorial Three factorial of course is three times two times one And then you have a four factorial on the bottom the four factorials will always cancel out here And then three times two cancel out with a six And so you get seven times five Which is 35 there are 35 possible committees That we could form by choosing three people from seven so we can compute these binomial coefficients using Uh these factorials now a way that's a little bit more convenient with respect to the binomial coefficient As we can to compute them recursively using a number array called pascal's triangle And i'm not talking about pascal from disney's movie tangled. It's not a chameleon here We're talking about the mathematician scientist basal pascal This is a recursive number a recursive triangle numbers that follows that is constructed by the following pattern So you start off with the number one then the next row is you're going to have a one and a one So you have two ones and every time you draw a new row it always starts and ends with ones So the first number and the last number is a one your bookends are ones So how do you construct all the subsequent numbers in the middle? So so you start off with a one you end with a one How do you get the two two is the sum of the two previous numbers above it? So one plus one is two So for the second for the third row here, you're going to then take one plus two, which is three And you're going to take two plus one, which is three Okay for the next row you're going to take one plus three, which is four You're going to take three plus three, which is six You're going to take three plus one, which is four for the next row You're going to take you start off with a one you're going to get one plus four, which is five You're going to get four plus six, which is 10 You're going to get six plus four, which is 10 and you're going to get four plus one, which is five right here One thing you should note with pascal's triangles that each row is a palindrome It's the same forward is backwards one five ten ten five one then I read that left to right or right to left You'll never know If we wanted to do the next row we can we can construct this recursively start off with a one One plus five is six five plus ten is 15 10 plus 10 is 20 and then it's going to repeat itself now 10 plus five is 15 five plus one is six And then you get a one the next row would be one one plus six is seven Six plus five is 21 Five 15 what did I say a six plus 15 is 21 15 plus 20 is 35 20 plus 15 is 35 15 plus six is 21 six plus one is seven and then you know one So that's just like we said here. We can keep on going going going going going when you look at pascal's triangle I should mention that since they always start off with a one you index the rows actually by the next number you see So this is actually considered the first row. This is the second row This is the third row and this is the fourth row etc So with referencing the rows you always go down that way Okay, uh, and so and so technically speaking This is the zero with row because there's no number right there And then as you go down the rows if you go through columns, this will always start off with the zero position This will be the first position the second position the third position and the fourth position So if you're out the nth row you're going to go from the zero spot zero one two three four five six seven you should always end with n The number you're on right now So we give an address to every number in this triangle and thus we're going to identify the numbers in the triangle with binomial coefficients So this right here is the zero zero position. That's zero choose zero This will be the one zero position. That's one choose zero. This is going to be the one one position That's one choose one. This will be the one or the two zero position. This is the two one position This is the two two position that actually means it's two two zero two choose one and two two two So what one can show is that this recursively constructed pascal's triangle gives us the binomial coefficients And so we can actually compute the binomial coefficients recursively Using pascal's triangle. So notice on the previous slide right We showed that seven choose three was equal to 35. Let's look on the triangle seven choose three We're going to get zero one two three. This number right here is seven choose three It's the seventh row third position seven choose three. That's not a coincidence We can compute the binomial coefficients using pascal's triangle. It's a very nice tool if you want to get them without the factorials