 We say that a plus b to power n is the power of a binomial, which we can expand using the binomial expansion. Well, all that's really saying is that a plus b to the n is the product of n factors of a plus b. Now, to find a term in the expansion of the power of a binomial, we need to do two things. First, determine which factors we need to choose, and then determine the number of ways we can choose those factors. For example, suppose I want to find the coefficient of x cubed y squared in the tenth power of x plus y. So again, it's helpful to remember x plus y to the tenth is the result of multiplying x plus y by itself ten times. So let's think about this. To get a term x cubed y squared, we have to choose x exactly three times and y exactly twice. But, since there are ten factors, we can't do that. And the thing to remember is you must choose a term from each of the factors. And so this is a pretty easy problem. The coefficient is zero. And in a kind and gentle universe, all binomial expansions would be this easy. We don't live in that universe. So let's find k and the coefficient of x cubed y to power k in the binomial expansion, x plus y to power ten. So to get x cubed y to k, well again, we have to choose x exactly three times. And remember, we have to choose a term from each factor. And since there are ten factors, that's our tenth power, we have to choose ten terms altogether. So if three of them are x, then ten minus three, seven of them must be y. And so we have to choose y exactly seven times. And so this means we can either choose three of the x factors out of the ten available, or seven of the y factors out of the ten available. We'll choose three from ten. So that's ten. Choose three is ten factorial divided by three factorial, seven factorial. And again, the easy way to remember this is that the denominator is going to be three factorial. That's one times two times three. And the numerator will start at ten and descend until we have three factors. So that's ten, nine, eight. And we'll compute. Now it's worth pointing out we get the same answer if we chose seven of the terms to be y. But in that case, our computation would be ten choose seven. That's ten factorial divided by seven factorial, three factorial. And again, we can write this out. Our denominator will be seven factorial. And our numerator would have seven factors starting with ten and going down. And numerically, we'd get exactly the same value. It's just that ten to seven is a little harder to compute. Now the important thing is who trusts me, I'm on the Internet and so everything I say must be the absolute truth. Because if it's not true, you couldn't put it online, right? Yeah, well with the binomial theorem we can find any term of any binomial expansion, but we could have done that before with some work and it's useful to compare the two results. Anytime somebody presents you with a new way of doing things, it's nice to check and make sure you're still getting the same results. So let's find the x term of 2x plus 5 squared by multiplying out and then by the binomial expansion. So if we multiply out, well we know how to do that. And we find our x term is 20x. In the binomial expansion, since each factor of our square is 2x plus 5 and we have two factors, to get the x term we'll need to choose 2x one time and 5 one time. And so the coefficient will be 2 choose 1, which will be 2, which is what we got the last time, well 0 is nothing, so 2 and 20 are the same thing. What's going on here? Well let's think about that a little bit more. The x term actually comes from choosing 2x and then 5 and this actually gives us 10x. We could also choose 5 then 2x, which gives us another 10x, so in fact we have two products of 10x. And so that's 10x plus 10x, well that's 20x. And so our actual term should be 20x. And here's the thing to notice that 2 that we got from the binomial coefficient is here as part of our product, but the rest of it comes from the factors themselves. And so remember in the expansion of a plus b to power n, the binomial coefficient must be multiplied by the appropriate powers of a and b. So for example let's say we want to find the x cube term in the expansion of x plus 2 to the seventh. So we have seven factors we have to choose. The x cube term has to come from choosing x three times and choosing 2 the remaining 7 minus 3 four times. So the binomial coefficient, well that'll be 7 choose 3. And since we've chosen x three times, we have an x to the third. And since we chose 2 four times, there's going to be a 2 to the fourth. We find 7 choose 3 and so the term itself will be, or let's find the x to the fifth term in the expansion of x minus 1 to the eighth. So again if we want to produce the x to the fifth term, we have to choose x five times and choose minus 1 the remaining 8 minus 5 three times. And so our term will be 8 choose 5 x to the fifth minus 1 to the third. And we can compute and our binomial could include a variable. So let's find the x cubed y squared term. So to get an x cubed y squared term we have to choose x three times minus 4 y two times. And so that's 5 choose 3 x to the third minus 4 y to the second. And so our term will be