International Mathematical Olympiad 1981 Problem 2

Let $1 \leq r \leq n$ and consider all subsets of $r$ elements of the set $\{1, 2, \ldots, n\}$. Each of these subsets has a smallest member. Let $F(n, r)$ denote the arithmetic mean of these smallest numbers; prove that $$F(n, r) = \frac{n + 1}{r + 1}.$$