Binomial Theorem

Find the coefficient of $x^{12}$ in

Find the constant term in

Since their sum is $2^n$, each is equal to $2^{n-1}$.

Differentiate

Then

Set $x=1$:

Differentiate twice:

Setting $x=1$:

Since

we obtain

For example, the coefficient of $x^2y^3z$ in

is

Because

the coefficients in a binomial expansion are symmetric.

For example,