For 8 $\star \star$ positive integers $r, n$ satisfying $1 \leqslant r \leqslant n$, find the arithmetic mean $f(r, n)$ of the smallest numbers in all $r$-element subsets of $\{1,2, \cdots, n\}$.
Parse $\{1,2, \cdots, n\}$ for a total of $\mathrm{C}_{n}^{r} \uparrow r$-element subsets, and find the sum of the smallest numbers in these $\mathrm{C}_{n}^{r}$ subsets.
For the $r$-element subset $A$ of $\{1,2, \cdots, n\}$, let its smallest number be $a$, and form $A \cup\{a-1\}, A \cup\{a-2\}, \cdots, A \cup\{0\}$ corresponding to it. Then, the smallest number in $A$ is the number of such sets, and the sum of all smallest numbers is the total number of all such sets, which is the total number of $(r+1)$-element subsets of the $(n+1)$-element set $\{0,1, \cdots, n\}$, i.e., $\mathrm{C}_{n+1}^{r+1}$.
Therefore, $f(r, n)=\frac{\mathrm{C}_{n+1}^{r+1}}{\mathrm{C}_{n}^{r}}=\frac{n+1}{r+1}$.