Combinatorics summation proof involving binomial coefficients

Question

Prove that

$$\sum_{k=0}^{n}\binom{n}{k}(-1)^kk^n = (-1)^nn!$$

Please can anyone help me out here?

Answer 1

Apply $\left(x \frac{d}{dx}\right)^n$ to both sides of the identity $(1-x)^n = \sum_{0 \leq k \leq n} \binom{n}{k} (-1)^k x^k $, then evaluate at $x=1$.