Prove a sum involving multinomial coefficients

Question

I need to prove that if n and m are positive integers, then

$$ \sum_{k_1+...+k_m}\binom{n}{k_1, ..., k_m}(-1)^{k_2+k_4+...+k_{2l}}$$

is equal to 0 if m =2l, and is equal to 1 if m = 2l + 1.

Answer 1

If $m$ is even, consider $$(\underbrace{1-1+1-1+\cdots+1-1}_{m/2 \,\,\,1's \text{ and } m/2 \,\,\, -1's})^n$$ If $m$ is odd, consider $$(\underbrace{1-1+1-1+\cdots+1-1+1}_{(m+1)/2 \,\,\,1's \text{ and } (m-1)/2 \,\,\, -1's})^n$$Look at the multinomial expansion and conclude the answer.