ILL 1990 Problem 64

Given an $m$ -element set $M$ and a $k$ -element subset $K \subset M$ . We call a function $f: K \to M$ has "path", if there exists an element $x_0 \in K$ such that $f(x_0) = x_0$ , or there exists a chain $x_0, x_1, \ldots, x_j = x_0 \in K$ such that $_xi = f(x_{i-1})$ for $i = 1, 2, \ldots, j$ . Find the number of functions $f: K \to M$ which have path.