## An identity

In the following problem we, as usual, define $0^0 = 1.$

Evaluate $\sum_{j=1}^n \binom{n}{j}(j-1)^{j-1}(n-j)^{n-j}$.

In case you want to see if your final answer is correct or not:

Spoiler:
The answer is $n^n - (n-1)^n$