College Algebra: Binomial Coefficients

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So, now as someone just told me that they pitied the person that would have to actually compute that triangle out to 101 spaces to see what all those coefficients are. So, happily it turns out there’s actually a formula where you can actually find each coefficient sort of ala carte, instead of actually going through that long triangle. So, I want to tell you about that in case you ever have to find a particular coefficient in some expansion. Which actually, believe it or not, can occur if you go on and do some more sort of number stuff.
Anyway, so the question is “How do you find the coefficients appearing in the binomial theorem, if you were to expand out a binomial?” Well, let me begin by introducing a new kind of function that takes an integer like seven or three or nine and spits out another number. It’s the factorial function. The factorial function, you take a number n and the way we denote the factorial is with an exclamation point. So, you read this n factorial or you could read it as n! Because, you see, it’s an exclamation point.
But what is the definition of this? Well, what you do to find out what this equals is you just take n and multiply it by n - 1 and multiply it by n - 2 and you keep multiplying it by successively less and less integers, subtracting one each time, until you get down to 2 x 1. That’s n factorial. So, it’s the product of all the natural numbers from n all the way down to 1. You just take the product of all those things. These are only defined for positive n. Negative three factorial makes no sense at all. Only for positive n, however, we do make a little convention. In fact, they call it a convention because I think they had a convention once and said, “Okay, we’ll do this.” The convention is that zero factorial we’ll just define to be one.
It doesn’t make sense to plug in zero factorial into this formula because I can’t take zero and subtracting one until I get to one. But we’ll just say, “All right, we all agree, anyone sees a zero factorial it’s just one.” It turns out a lot of the formulas work out better that way. So, that’s fine. But, let’s take a look at, for example, four factorial. What is that? Well, that would just be 4 x 3 x 2 x 1. I just take all the numbers less than 4, the integers, down to 1 and multiply them together. So, I’d see 12 and 2 is 24.
Now, one thing that you can see pretty quickly is the factorial just grows like the dickens. So, I mean, 4 factorial is already 24. You can figure out what 5 factorial is. If you think about it would just be 5 times 4 factorial. Do you see that? That’s sort of a fun thing to just notice. That if I took 5 factorial that would equal 5 x 4 x 3 x 2 x 1. But that’s exactly 4 factorial. So, 5 factorial is just 5 times 4 factorial. You can sort of peel one off and you put the 5 there. You can take 5 times that and what would that be? Well, that would be a zero. Carry the two. So it’d look like it would be 120. Huge number and that’s 5 factorial. Imagine what 107 factorial is. I don’t even want to go into it with you. I don’t have enough time.
So, this is a function that if you put in a number it spits out this kind of thing. Well, just given that I can now give you a formula for the coefficients for the binomial expansion. It turns out these are called binomial coefficients and it’s really sort of a neat symbol. We write the binomial coefficient of n m. We sort of set it up like a parenthesis thing. It looks like a fraction almost, doesn’t it? But there’s no line here. It’s just parenthesis and I say n here and m here and this would be read the binomial coefficient n m or sometimes this is said n choose m. We’ll actually see this symbol and it’s meaning more in a couple of sections if you take a look at the combinations section. In fact, this is actually a combination and we’ll talk about that a little bit later.
But anyway, it’s just a complicated looking thing, but all it is, is a particular number. Here’s the number. You just take n factorial, I told you what that is, and divide it by m factorial multiplied by n - m factorial. So, it’s sort of a fraction. The first thing to notice is that for this to make sense, n has to be bigger than or equal to m. If m were bigger then I’d have a negative number here and that’s not allowed. So, this is only when n is greater than or equal to m. That’s how I define this.
Well, it turns out that with just definitions--let’s just try an example here just to see if you can sort of make some sense out of this. So, let’s find the binomial coefficient of 4 and a 2 down here. What does the recipe say? It says I take 4 factorial and I divide that by 2 factorial. Then 4 - 2, which is another 2 factorial. So, what does that equal? Well, the top equals 4 x 3 x 2 x 1 and the bottom is 2 x 1 x 2 x 1. So, those ones have no effect. You’ll notice the 2 x 2 actually produces a 4. So, in fact, that cancels with this 4 here. So, what does this equal? This just equals 6.
That’s a little bit cool I think. Because look, the formula is a fraction and it turns out in this particular example that fraction turns out to be just an integer. It turns out to be just six. And you know what? That is not a freak of nature. This number will always be an integer. Even though it looks like a fraction, you’ll always have sort of complete cancellation like that at the bottom. You’ll always be left with an integer. So, these numbers are actually integers. Even though it’s not obvious. They look like fractions, but if you try many, many examples you’ll start to see that these always seem to be integers and you can actually prove that if you wanted to.
Anyway, that’s how you compute this number and what does this number have to do with anything? Well, it turns out that this number is what allows us to give the binomial formula. So, let me see if I can just put this up here. Now, I can tell you what the binomial theorem says. The binomial theorem says the following. It says that if you want to expand out x + y raised to some power, let’s say the n power, then here’s what you do. That’s going to equal xn plus and now there’s going to be some coefficient and it’s the binomial coefficient. It’s the one that’s n choose 1. So, you evaluate this with an n here and a 1 here. Then you multiply that by xn-1. I deduct one exponent from here and then I stick on a y. Then plus, what’s the next term?
Well, it’s the next binomial coefficient, which is n choose 2. So, I just go back into here. Put an n in here and put a 2 in here and compute that number. That’s the coefficient. Then I have xn-2 times y² and I keep going. Plus the next coefficient. You can probably guess it, would be n choose 3. So, I go back to this formula and put a 3 in for m. Then I deduct one on the exponent on x. So, I have n, xn-3, y and I increase the y, y³. Notice how the sum of the exponents is always n, n - 3 + 3 is n. We always have a sum of n if you add the exponents up, if I’m expanding something to the nth power.
Then we just keep going and keep doing this until we get down to the end. The penultimate entry would be n choose n - 1. You see how I keep increasing, one, two, three and I keep going. Here’s the penultimate one, n - 1. This would just be to the times x, times yn-1. The very last entry would be n choose n times y to the n. So, in fact, these give you the binomial coefficients when you expand out a binomial to the nth power those numbers that should go in there, in fact, these are just a line in Pascal’s triangle. So, this is a formula for Pascal’s triangle if you will. You just plug in and you can find these numbers.
So, let me sort of illustrate all this theme with an example. Suppose someone said to you, I’m going to try to do sort of a semi practical one. Suppose someone says to you “Well, actually multiply out 2A - B all to the fifth power.” Well, you could trudge through and multiply that out all five times. 2A - B times 2A - B times 2A - B times 2A - B times 2A - b. You could do that. That would work. Another way is to try to use these binomial theorem ideas. I could write this now as 2a, there’s that piece, plus -b, there’s that piece all to the fifth. You see how I now have it in the shape of the binomial theorem. It’s something plus something raised to a power.
So, I could actually use the binomial theorem thing, which says I first have to find the coefficients. So, how do I do that? Well, you’re either use the binomial coefficients and do all the factorial stuff, which is fine, or since this number is sufficiently small me might just sort of zip through Pascal’s triangle really fast. Let’s do Pascal’s triangle. So, we’ve got a one and then we’ve got a one, one. Some people don’t start with the one, by the way. I don’t know. It’s sort of good to make it sort of a real triangle. But this is the first level, really. Then I have one, two one. That’s if we square. One, three, three, one. That’s if we cube. One, four, six, four, one. That’s if we raise things to the fourth power. Then if we raised things to the fifth power, we’d see one, five, ten, ten, five, and one. So, since that’s the fifth level, these are the coefficients in order. Don’t get confused and think that’s a level. I just put that there to sort of make up the teepee. If you don’t like that, just don’t write it in there.
This is the binomial, x + y, and this (x + y)², x² + 2xy + y². So, there are the coefficients. So, what do I do? I just ought to be very careful with how I report the news and I should be fine. So, first I take this coefficient, which is just a one and I take this term to the fifth power. So, what I would see is 2A to the fifth power. Then A +5, because that coefficients a 5. Then I have 2A to the fourth power and now I put in one of these. So, times -B to the first power. Then what do I see? Well, then I see a 10. So, I have +10 and then I take the 2A to the third power and I bump up the -B and make it A -b². Then I take another +10 and I have +10. Now I do what?
I decrease the exponent on the 2a. So, I see just 2a². Then I up the exponent on the -b. So, I have -b³. Notice, by the way, at any stage if you add the exponents I get 5, 2 + 3 is 5, 3 + 2 is 5, 4 + 1 is 5, 5 + 0 is 5. Always 5. That will always happen. Then I have the penultimate term, which is a 5 coefficient. Then I have a 2A to the first power times -B to the fourth. And the very, very last term is going to be -B to the fifth power with the coefficient of a one.
Okay, so that’s the answer and now of course you might want to expand that out a little bit. You could write this as for example, 25A5 plus and you could write 5 x 24 times A4, but there’s a negative sign there. So, I could put a B here, but then change this sign to negative. Just pulling the negative out in front. Here when I take the -B and square it, it becomes a positive again. So, I just see +10 x 2³. A³B² and then you can add plus, well -B³ is going to give me a negative B³. So, I could actually make this a minus. So, I’d have a -10 x 2² x A²B³ and this term here is going to be -B4. That minus sign will therefore go away, because I have -1 to an even power.
So, I just would see + 5 times 2AB4 and then -B5 gives me a minus sign out in front, just B5. Then if you want to you could put down what 25 is, 24 and multiply it by the 5, 2³, 8, multiply it by 10 and get 80 and so on. If you do that you could see it all there on one line, nice and pretty. In fact, you’ve now expanded this thing out into all its binomials. All I used there was just the binomial theorem in order to expand everything out. Okay. Try some of these and it turns out that these things will be handy when you want to expand something to a really high power. All right, try the binomial theorem and make it your friend. See you soon.