Binomial thereom to figure out coefficents

Question

Use the binomial theorem to find the coefficient of $x^8y^5$ in $(x + y)^{15}$

My textbook shows how to do this looking at the coefficents of Pascal's triangle but, I know theres another way using combinations. Could someone explain to me how I would figure this out without using the triangle?

Answer 1

It is:

$$(x+y)^n=\sum_{k=0}^n \binom{n}{k} x^ky^{n-k}$$

$$(x+y)^{15}=\sum_{k=0}^{15} \binom{15}{k} x^k y^{15-k}= y^{15}+ \dots+ \binom{15}{8}x^8y^{7} + \dots + x^{15}$$

So,if we add the powers of $x$ and $y$,the result must be $15$.

At $x^8y^{5}$,if we add the powers,the result is $5+8=13 \neq 15$,so the coefficient is equal to $0$.