Question 8
Let $p$ be a permutation of the set $S_{n} = \{1, 2, \ldots, n\}$. An element $j \in S_{n}$ is called a fixed point of $p$ if $p(j) = j$. Let $f_{n}$ be the number of permutations having no fixed points, and $g_{n}$ be the number with exactly one fixed point. Show that $\left|f_{n} - g_{n}\right| = 1$.