HMMT 2003 C8

For any subset $S \subseteq\{1,2, \ldots, 15\}$, a number $n$ is called an "anchor" for $S$ if $n$ and $n+|S|$ are both members of $S$, where $|S|$ denotes the number of members of $S$. Find the average number of anchors over all possible subsets $S \subseteq\{1,2, \ldots, 15\}$.