How to calculate a combinatorial sum

HINT: Start with $$f(x)=(1+x)^n=\sum_{k=0}^n\binom{n}{k}x^k.$$ Differentiate to show that $f'(-1)=0$. Differentiate again to show that $f''(-1)=0$. Keep going (inductively) until the $(n-1)$-th derivative.

For a combinatorial proof, note that both sides count the number of surjections of an $(n-1)$-set onto an $n$-set. The RHS is obvious, and the LHS is the inclusion-exclusion formula, where the $n$ properties to be avoided are that $i$ is not in the image of the function. More generally, this proof shows that the identity holds if you replace the power $n-1$ with any number $<n$.