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 6 of 6: Divide by the number of subsets

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

Detailed analysis

Dividing the total $\binom{n+1}{r+1}$ from Step 5 by the total number of subsets $\binom{n}{r}$ gives the mean $F(n,r)$ , which simplifies to $\frac{n+1}{r+1}$ exactly as in Solution 1.