Binomial Identity/Coefficient Question

Question

So I know that the coefficient of $[x^n]$ is computed by using:

$\left( \sum_{j=0}^n a_j b_{n-j} \right)$ = $[x^n]A(x)B(x)$

How is this formula used to make computations, for example, how do I compute:

$[x^7](1+x)^{20}$ ?

Answer 1

You can take $A(x)=(1+x)^{10}$ which implies $B(x)=A(x)=(1+x)^{10}$.

Added: From the identity, I gave in the note, you can see that

$$ a_j = {10\choose j},\quad b_j=a_j={10\choose j}, $$

which gives

$$ [x^7]A^2(x) = \sum_{j=0}^{7} {10\choose j} {10\choose 7-j}=\dots\,. $$

Note: You need the identity

$$ (1+x)^m = \sum_{i=0}^{m} {m\choose i}x^i. $$