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Collection of pairwise mutually disjoint p-element subsets

Source: India tst 2006 p5

June 27, 2012
inductionnumber theorycombinatorics proposedcombinatorics

Problem Statement

Let pp be a prime number and let XX be a finite set containing at least pp elements. A collection of pairwise mutually disjoint pp-element subsets of XX is called a pp-family. (In particular, the empty collection is a pp-family.) Let AA(respectively, BB) denote the number of pp-families having an even (respectively, odd) number of pp-element subsets of XX. Prove that AA and BB differ by a multiple of pp.
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