 Sometimes when you're multiplying out polynomials, you don't necessarily need every coefficient. There might just be one or two coefficients that are actually important to you. And the binomial coefficients can also be helpful for finding that, like what if we wanna find the coefficient of y to the eighth in the expansion of two y plus three to the 10th? We could do all of the powers, we could, but to get y to the eighth, how do you form a y to the eighth? To form a y to the eighth, you're gonna take two y to the eighth power, then you're gonna have to take three to the second power there, because the combined total, eight and two, is gonna have to add up to be 10. So in that binomial expansion, you're gonna take eight y's and then two threes, and of course you have to grab two eight times as well. But you're also gonna have the binomial coefficient of 10 choose eight, which you could also do choose two, it's the same thing, it really doesn't matter. So we wanna simplify this thing. 10 choose eight means 10 factorial, 10 factorial over two factorial and eight factorial. You're gonna get two to the eighth times y to the eighth, and you're also gonna get three squared. And so let's simplify these things. To eight factorial is gonna go into 10 factorial, leaving eight 10 times nine on top. Two factorial is just two itself. Three squared is a nine. Two to the eighth, what is that gonna be? So we think of powers of two, so you're gonna get two, four, eight, 16, 32, that's the fifth power. 64 is the sixth power, then 128, 256. I mean, you can also use a calculator here. You're gonna get 256 times y to the eighth right here. Let's simplify this where we can. Let's see, two does go into 256 right here, leaving 128 behind. If I take 10 times 128, that's pretty nice. You're gonna get 1280 times 8199 to the eighth right there. And then I'm not gonna pretend like I'm gonna be able to do this one in my head. 80, two digit multiplication, no way, Jose. Multiply those together. You're gonna get 103,680 times y to the eighth. That's what the y to the eighth term's gonna look like in that expansion. Therefore, the coefficient is 103,680. We don't need all of the expansion if we just wanna find the coefficient of y to the eighth right here. Let's look at another example of this. Let's find the sixth term, the sixth term of x plus two to the ninth when written in descending order. Why don't we buy the sixth term, right? So the sixth term would be like, well, the first term is gonna have a, you're gonna have an x to the ninth, then there's gonna be some x to the eighth, x to the seventh, x to the sixth, x to the fifth, x to the fourth, right? There's gonna be some coefficients here. We just don't know what they are. I mean, we can compute them. So it's like, one, two, three, four, five, six. So we're trying to figure out what's the coefficient? What's the coefficient of x to the fourth? That's really what they're asking in this example right here. So it's similar to what we did previously. So how does one get an x to the fourth? Well, you're gonna grab x four times, which means you're gonna grab two five times, four plus five is equal to nine. And then you're gonna take nine, choose four, or you could do nine, choose five. Am I grabbing four? Am I grabbing x four times or am I grabbing two five times? It really doesn't matter. It's the same number. And so you're gonna get nine factorial over four factorial and five factorial. You're gonna get an x to the fourth. You also get this 32, two to the fifth right there. And so five factorial is gonna go into nine factorial, leaving a nine times, eight times, seven times, six, four of the terms over four factorial is gonna get four times, three times, two. Three and two comes together as a six. So that will cancel out. Four goes into eight, two times. That leaves also, there's still the 32x to the fourth right there. And so there we have, we have nine times, two times, seven times, 32 times x to the fourth. And so when we multiply all those together, let's see, two times, 32 is a 64. Nine times seven is 63. And then 64 times 63, that's equal to 4032x to the fourth. So the coefficient of the sixth term will be 4032. So if we need to know just one of the coefficients, we can figure that out without having to multiply everything out. And there, that gives us a nice little application of the binomial theorem.