In the following problem we, as usual, define $\displaystyle 0^0 = 1.$ Evaluate $\displaystyle \sum_{j=1}^n \binom{n}{j}(j-1)^{j-1}(n-j)^{n-j}$. In case you want to see if your final answer is correct or not: Spoiler: The answer is $\displaystyle n^n - (n-1)^n$
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