ISL 1989 Problem 95

Let $n$ be a positive integer, $X = \{1, 2, \ldots , n\},$ and $k$ a positive integer such that $\frac{n}{2} \leq k \leq n.$ Determine, with proof, the number of all functions $f : X \mapsto X$ that satisfy the following conditions:

(i) $f^2 = f;$ (ii) the number of elements in the image of $f$ is $k;$ (iii) for each $y$ in the image of $f,$ the number of all points $x \in X$ such that $f(x)=y$ is at most $2.$