CMO 1982 Problem 4

Let $p$ be a permutation of the set $S_n = \{1, 2, \dots, n\}$ . 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 $|f_n - g_n| = 1$ .