Computing the sum $\sum_{k=0}^{n} \binom{n}{k}(-1)^k(n-k)^n$ I need to compute $S_{n} = \sum_{k=0}^{n} \binom{n}{k}(-1)^k(n-k)^n$.
$S_{2} = 2, S_{3} = 6,S_{4}=24$ therefore i think answer is $S_{n} = n!$.
And i have got $S_{n}=(\Delta^n(n-x)^n)(0)$.
But it gives no benefit to me..

Hint
It's precisely the cardinality of the set $$\left\{f:\{1,2,...,n\}\to \{1,2,...,n\}\mid f\ \text{ is onto (or surjective)}\right\}$$
Moreover, you know that a function $f:\{1,...,n\}\to\{1,...,n\}$ is onto if and only if is injective and thus bijective. I let you continue.
