Question 14
Let $n\gt 1$ be an integer. There are $n$ boxes in a row, and we have $n+1$ identical stones. A distribution is a way of distributing the stones over the boxes, where each stone is in exactly one box. We say that two such distributions are at a stone's throw from each other if we can obtain one distribution from the other by moving exactly one stone to a different box. The sociability of a distribution $a$ is defined as the number of distributions that are at a stone's throw from $a$. Determine the average sociability over all possible distributions.
(Two distributions are the same if the $i$-th box of both rows contains the same number of stones, for all $i \in \{1,2, \ldots, n\}$.)
(Two distributions are the same if the $i$-th box of both rows contains the same number of stones, for all $i \in \{1,2, \ldots, n\}$.)