Question 16
(FRG 3) (IMO1) Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k)$. Prove:
(a) $\sum_{k=0}^n kp_n(k)=n!$;
(b) $\sum_{k=0}^n(k-1)^2p_n(k)=n!$.
(a) $\sum_{k=0}^n kp_n(k)=n!$;
(b) $\sum_{k=0}^n(k-1)^2p_n(k)=n!$.