TITLE: Show this equality (The factorial as an alternate sum with binomial coefficients).
QUESTION [10 upvotes]: Why does the following equality hold?
$$n!=\sum_{k=1}^n (-1)^{n-k} \binom{n}{k} k^n$$

REPLY [5 votes]: 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 $x=1$.