Question 13
Let $p_{n}(k)$ be the number of permutations of the set $\{1,2, \ldots, n\}$, $n \geq 1$, which have exactly $k$ fixed points. Demonstrate that $$
\sum_{k=0}^{n} k p_{n}(k) = n!
$$ (Note: A $f$ permutation of a $S$ set is a bijective application of $S$ on itself. An element $i$ $S$ is called the permutation fixed point $f$ if $f(i) = i$.)
\sum_{k=0}^{n} k p_{n}(k) = n!
$$ (Note: A $f$ permutation of a $S$ set is a bijective application of $S$ on itself. An element $i$ $S$ is called the permutation fixed point $f$ if $f(i) = i$.)