>> Now, let's talk a bit about the coefficient generally in a binomial expansion. It turns out that that is what the binomial theorem is all about. In the X plus Y to the 12th expansion, find the coefficient of the term involving X to the 7th, Y to the 5th. And the binomial theorem allows us to make that calculation. Now, I'm using a specific example here. We can generalize this in a little bit, but the specific example is this specific expansion. All right, now, to generalize it, we would say it's the expansion of X plus Y to the N power, you know, something like that. But this is very specific. Now, by the binomial theorem, that coefficient is given by this notation and this calculation. The notation here is read. This is the combination of 12 things, taken 5 at a time. And another way to show the exact same notation is to do this. This says the same thing. And the calculation associated with this is 12 factorial over 12 minus five factorial times 5 factorial. Now, let's see where all of this comes from and where we're going with this and how we can manipulate this idea. By the way, the notion of combinations is heavily involved in the study of probability. And it's something that we'll get involved with at another time. But just for the time being, let's just understand that it's a particular notation, and this is its correspondence. All right, now, the 12 appearing here corresponds with the exponent on the X plus Y, so that's where that comes from. And it turns out that the five here happens to be the exponent on the Y. You're saying the exponent on Y, the second part of the binomial expansion, you see. Okay, now, let's make the correspondence between these two. We have 12 factorial. That corresponds here with the numerator, you see? And then in the denominator we have a couple of items. One of the items is the difference of these two factorial, and the other item is this second rascal factorial. Okay, so all of that is going on. Now, it's only a matter of experience that allows us to remember this and understand it well. I don't recommend that we go and try to memorize this. We'll get enough practice with it so that the memory will be very efficient very shortly. All right, let's make the calculation. Let's see, we have 12 factorial is 12 times 11 times 10 times 9 times 8 times 7 -- and I'm gonna stop right there. I know that there are some more factors, but I'm seeing that in the denominator, I have 12 minus 5, which is 7 factorial. So I have 7 factorial times 5 factorial. Now, 5 factorial, I'll go ahead and expand as 5 times 4 times 3 times 2, and I could write the one here, but I don't need to. Now, I stopped here at 7 factorial because I recognize this and all the factors involved there will take out all of the factors involved here. And now, we're looking for other cancellation opportunities. I see 5 times 2 is 10 will cancel with the 10 upstairs. I see 4 times three, twelve. And those factors will cancel with the 12 upstairs. So we just multiply 11 times 9 times 8, and we find the value here to be 792.