Here's the question:
Prove that for all positive integers k ≤ n,
$\sum_{i=0}^{k} {n \choose i} (-1)^i = {n -1 \choose k}(-1)^k$
So far, I've noticed that you can change $ {n \choose i}$ into ${n - 1 \choose i} + {n -1 \choose i - 1}$, which gets me closer to where I want to be, but beyond that I'm stuck.