TH-TMO 2005 Problem 8

For each subset $T$ of $S = \{1, 2, ... , 7\}$ , the result $r(T)$ of T is computed as follows: the elements of $T$ are written, largest to smallest, and alternating signs $(+, -)$ starting with $+$ are put in front of each number. The value of the resulting expression is $r(T)$ . (For example, for $T =\{2, 4, 7\}$ , we have $r(T) = +7 - 4 + 2 = 5$ .) Compute the sum of $r(T)$ as $T$ ranges over all subsets of $S$ .