# Problem 201: Subsets with a unique sum
For any set A of numbers, let sum(A) be the sum of the elements of A.
Consider the set B = {1,3,6,8,10,11}. There are 20 subsets of B
containing three elements, and their sums are: sum({1,3,6}) = 10,
sum({1,3,8}) = 12, sum({1,3,10}) = 14, sum({1,3,11}) = 15, sum({1,6,8})
= 15, sum({1,6,10}) = 17, sum({1,6,11}) = 18, sum({1,8,10}) = 19,
sum({1,8,11}) = 20, sum({1,10,11}) = 22, sum({3,6,8}) = 17,
sum({3,6,10}) = 19, sum({3,6,11}) = 20, sum({3,8,10}) = 21,
sum({3,8,11}) = 22, sum({3,10,11}) = 24, sum({6,8,10}) = 24,
sum({6,8,11}) = 25, sum({6,10,11}) = 27, sum({8,10,11}) = 29. Some of
these sums occur more than once, others are unique. For a set A, let
U(A,k) be the set of unique sums of k-element subsets of A, in our
example we find U(B,3) = {10,12,14,18,21,25,27,29} and sum(U(B,3)) =
156. Now consider the 100-element set S = {12, 22, ... , 1002}. S has
100891344545564193334812497256 50-element subsets. Determine the sum of
all integers which are the sum of exactly one of the 50-element subsets
of S, i.e. find sum(U(S,50)).