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Summing up cardinality of intersection of sets with union=S

Source: RMO (Mumbai Region) 2015 P6

December 6, 2015
combinatoricscombinatorics proposedEnumerative Combinatoricscounting

Problem Statement

Let S={1,2,⋯ ,n}S=\{1,2,\cdots, n\} and let TT be the set of all ordered triples of subsets of SS, say (A1,A2,A3)(A_1, A_2, A_3), such that A1∪A2∪A3=SA_1\cup A_2\cup A_3=S. Determine, in terms of nn, ∑(A1,A2,A3)∈T∣A1∩A2∩A3∣ \sum_{(A_1,A_2,A_3)\in T}|A_1\cap A_2\cap A_3|
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