Working with binomial coefficient $\sum_{k=0}^n (-1)^k \binom nk=0$

Question

I'm sure there's an identity or some known "trick" to solve this but I couldn't find it (Looked in Wikipedia, Wolfram and here):

$$ \sum_{k=0}^n (-1)^k{n \choose k} =0 $$

I want to know how this can be proven. I tried induction but I'm not sure what to do with ${{n+1}\choose k}$.

Thanks!

Answer 1

$$ (x+y)^n = \sum_{k=0}^n\left(\matrix{n\\k}\right)x^ky^{n-k} $$ set $x=-1$ and $y=1$