ELMO SL 2010 C1

For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$ , let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$ . [list=1] [] Given $n$ , what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$ ? [] Given $n$ , what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$ ?[/list]

Brian Hamrick.