MATH190 presentation next week

I am here to post my answer (numerical and method only) for my teammate, you can use it to check. If our answers are different,  please discuss with me.

Problem (1990 Austrian-Polish Math Competition)
Let $n>1$ be an integer and let $f_1,f_2,\dots, f_{n!}$ be the $n!$ permutations of $1,2,\dots,n$ (each $f_i$ is a bijective function from $\{1,2,\dots,n\}$ to itself).  For each permutation $f_i$, let us define $\displaystyle S(f_i)=\sum_{k=1}^n|f_i(k)-k|$. Find $\displaystyle \frac{1}{n!}\sum_{i=1}^{n!}S(f_i)$.

My numerical answer. $\displaystyle \frac{n^2-1}{3}$.

Next question I just use Jensen's inequality once.