2. Let $1 \leqslant r \leqslant n$. Then the arithmetic mean of the largest numbers in the $r$-element subsets of the set $M=\{1,2, \cdots, n\}$ is $\qquad$ .
2. $\frac{r(n+1)}{r+1}$.

The number of subsets of $M$ containing $r$ elements is $\mathrm{C}_{n}^{r}$. The number of subsets of $M$ containing $r$ elements with the largest number being the positive integer $k$ is $\mathrm{C}_{k-1}^{r-1}$. Therefore, the arithmetic mean of the largest numbers in the subsets of $M$ containing $r$ elements is
$$
\begin{array}{l}
\frac{1}{\mathrm{C}_{n}^{r}}\left[r \mathrm{C}_{r-1}^{r-1}+(r+1) \mathrm{C}_{r}^{r-1}+\cdots+n \mathrm{C}_{n-1}^{r-1}\right] \\
=\frac{r}{\mathrm{C}_{n}^{r}}\left(\mathrm{C}_{r}^{r}+\mathrm{C}_{r+1}^{r}+\cdots+\mathrm{C}_{n}^{r}\right) \\
=\frac{r \mathrm{C}_{n+1}^{r+1}}{\mathrm{C}_{n}^{r}}=\frac{r(n+1)}{r+1} .
\end{array}
$$