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Find the sum of some sums of a set

Source:

November 13, 2005
AMCAIMEprobabilityalgebrabinomial theorem

Problem Statement

For {1,2,3,…,n}\{1, 2, 3, \dots, n\} and each of its nonempty subsets a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for {1,2,4,6,9}\{1, 2, 4, 6, 9\} is 9−6+4−2+1=69 - 6 + 4 - 2 + 1 = 6 and for {5}\{5\} it is simply 5.) Find the sum of all such alternating sums for n=7n = 7.
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