data/problems/201.yml
---
:id: 201
:name: Subsets with a unique sum
:url: https://projecteuler.net/problem=201
:content: "For any set A of numbers, let sum(A) be the sum of the elements of A. \nConsider
the set B = {1,3,6,8,10,11}. \n There are 20 subsets of B containing three elements,
and their sums are:\n\nsum({1,3,6}) = 10, \nsum({1,3,8}) = 12, \nsum({1,3,10})
= 14, \nsum({1,3,11}) = 15, \nsum({1,6,8}) = 15, \nsum({1,6,10}) = 17, \nsum({1,6,11})
= 18, \nsum({1,8,10}) = 19, \nsum({1,8,11}) = 20, \nsum({1,10,11}) = 22, \nsum({3,6,8})
= 17, \nsum({3,6,10}) = 19, \nsum({3,6,11}) = 20, \nsum({3,8,10}) = 21, \nsum({3,8,11})
= 22, \nsum({3,10,11}) = 24, \nsum({6,8,10}) = 24, \nsum({6,8,11}) = 25, \nsum({6,10,11})
= 27, \nsum({8,10,11}) = 29.\n\nSome of these sums occur more than once, others
are unique. \nFor 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.\n\nNow consider the 100-element set S = {1<sup>2</sup>, 2<sup>2</sup>, ...
, 100<sup>2</sup>}. \nS has 100891344545564193334812497256 50-element subsets.\n\nDetermine
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)).\n\n"