For every nonempty subset $X$ of $M=\{1,2,\ldots,2000\}$, $a_X$ denotes the sum of the minimum and maximum element of $X$. Compute the arithmetic mean of the numbers $a_X$ when $X$ goes over all nonempty subsets $X$ of $M$.
Source: Moldova 2000 Grade 10 P3
Tags: combinatorics
For every nonempty subset $X$ of $M=\{1,2,\ldots,2000\}$, $a_X$ denotes the sum of the minimum and maximum element of $X$. Compute the arithmetic mean of the numbers $a_X$ when $X$ goes over all nonempty subsets $X$ of $M$.