# Combinatorics summation proof involving binomial coefficients [duplicate]

Prove that

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

Please can anyone help me out here?

## marked as duplicate by Théophile, Dragonemperor42, N. F. Taussig, Claude Leibovici, JonMark PerryMar 26 '17 at 8:05

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## 1 Answer

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 $x=1$.