Finding Coefficients Quickly in the TMUA

追加: 2026-05-22

[00:00:00]Today, I've got a nice and quick problem from the TMUA paper 2 2019. Oh, no, sorry, 2022. Um it's a nice little problem with a nice little trick. This one's one of those questions where there's probably an easy way and a a sorry, a short way and a long way to solve this. I imagine a few people would have done this the long way. So, for those people, this video is for you. How can you do this the short way? So, the long way is probably just expand this all out. We want to find the coefficient of x to the 5 in this expansion. So, you could think to expand that and then work out what that is. And that's kind of what we're doing, we're just doing it sensibly. So, 1 + x to the 5 * this thing here. Well, what is this? This is just 1 + x + x squared + x cubed + x to the 4 + x to the 5. Now, you could kind of expand this out and then you know, you compare there. Or we can be kind of smart. We know this is a polynomial of degree 5, so is this guy here. And all the coefficients here are just 1. So, to work out the coefficient of x to the 5 in the whole function, we just got to think, well, how can we take something from this guy and something from this guy, multiply them together, and get an x to the 5? Well, we know that this left bracket here is going to be like 1 + 5 choose 1 x + 5 choose 2 x squared and so on up to x to the 5.

[00:01:07]So, what we could do, let me change my pen color here, is take the 1 from here and the x to the 5 from there.

[00:01:12]Multiplying those together gives us one lot of x to the 5.

[00:01:16]We can choose the five the five choose one from there and then the the x to the 4 from here cuz 5 choose 1 * x * x to the 4 gives us an x to the 5.

[00:01:24]That's going to be 5 choose 1.

[00:01:25]Similarly, the 5 choose 2 x squared * the x cubed that gives us another x to the 5. And we can kind of see the pattern here. Um we're just going to get something like this.

[00:01:36]And in fact, this 1 at the start we can think of as 5 choose 0. And we've got the sum of these binomial coefficients.

[00:01:41]And again, you could try manually work these out one by one, but a strong mathematician should know that this will just be 2 to the 5. There's a few ways that you can kind of see this. There's a bunch of ways you can think about this.

[00:01:52]Um you can just think about like 1 + x to the 5. These are just the you know, if I write this out, this is 1 + 5 choose 1 x. I've written it out over here. This is just 1 + x to the 5.

[00:02:03]And the idea is if you just plug in 1 on both sides, on the left-hand side you get 2 to the 5, and on the right-hand side you get 1 + 5 choose 1 + 5 choose 2, and so on up to 5 choose 5. And that there is exactly why this is 2 to the 5.

[00:02:16]So, our answer here is 32.

[00:02:18]A pretty standard result that you should know, and if you're also preparing for the interviews, you should also be able to explain where this result like 1 + x to the n, oh sorry, 1 + n choose 1 + n choose 2 + blah blah blah + n choose n = 2 to the n.