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TOT 238 1989 Autumn A S2 sum of squares of products of subsets 1,2,..,N

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March 12, 2021
combinatoricsfactorialSubsets

Problem Statement

Consider all the possible subsets of the set {1,2,...,N}\{1,2,..., N\} which do not contain any consecutive numbers. Prove that the sum of the squares of the products of the numbers in these subsets is (N+1)!āˆ’1(N + 1)! - 1.
(Based on idea of R.P. Stanley)
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