Prove that $\sum\limits_{k=0}^n(-1)^k \binom{n}{k}k^n=(-1)^nn!$

I was trying to find the general term of the sequence $$a_n=\sum\limits_{k=0}^n(-1)^k \binom{n}{k}k^n$$ .

When I type the following in MMA

Sum[(-1)^k Binomial[n, k] k^n, {k, 0, n}] 

I get a simple expression $$(-1)^n n!$$.

But I find it difficult to prove this equality by induction.

So I wonder if there exists some ways to get a step-by-step evaluation of the function Sum.

Note that the methods mentioned here are not working in this case.