I suppose that the answer should be given in a closed form. I can see that it can be written shorter using the sum symbol ($\displaystyle \sum_{n=0}^{k}{k\choose n}(k-n)^k(-1)^n$)...
That happens to count the number of surjections from a set of k to a set of k objects.
Or another way it counts surjections from a finite set of k objects onto itself. Those happen to be permutations.
How ways can we permute k objects. That is your answer.