2002 AIME II Problems/Problem 9
Let be the set Let be the number of sets of two non-empty disjoint subsets of . (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when is divided by .
Let the two disjoint subsets be and , and let . For each , either , , or . So there are ways to organize the elements of into disjoint , , and .
However, there are ways to organize the elements of such that and , and there are ways to organize the elements of such that and . But, the combination such that and is counted twice.
Thus, there are ordered pairs of sets . But since the question asks for the number of unordered sets , .
Let and be the disjoint subsets. If has elements, then the number of elements of can be any positive integer number less than or equal to . So
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