Let p_{n}(k) of permutations of an n-element set that leave exactly k points fixed. Then

\displaystyle\sum_{k=0}^{n}p_{n}(k) =n!
\displaystyle\sum_{k=0}^{n}kp_{n}(k) =n!
\displaystyle\sum_{k=0}^{n}(k-1)^{2}p_{n}(k) =n!

I placed the proof on a separate page.