Competition problem — Combinatorics

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Combinatorics Difficulty 4.9 AIME Find the answer

Find the number of subsets $S$ of $\{1,2, \ldots 63\}$ the sum of whose elements is 2008.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Note that $1+2+\cdots+63=2016$ . So the problem is equivalent to finding the number of subsets of $\{1,2, \cdots 63\}$ whose sum of elements is 8. We can count this by hand: $\{8\},\{1,7\},\{2,6\}$ , $\{3,5\},\{1,2,5\},\{1,3,4\}$ .

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