Problem 2

Let

n

and

r

be integers with

1 \le r \le n

, and consider all

\binom{n}{r}

subsets of

r

elements of the set

\{1,2,\dots,n\}

. Each such subset has a smallest element. Let

F(n,r)

denote the arithmetic mean of these smallest elements. Prove that

F(n,r) = \frac{n+1}{r+1}.

Step 4 of 5: Form the arithmetic mean

F(n,r) = \frac{\sum_{S}\min(S)}{\binom{n}{r}} = \frac{\binom{n+1}{r+1}}{\binom{n}{r}}

Detailed analysis

There are $\binom{n}{r}$ subsets in total, so dividing the total from Step 3 by $\binom{n}{r}$ gives the mean $F(n,r)$ of the smallest elements.