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E(a)is the number of elements of $X_1\cup ... \cup X_n, \sum_{a\in S} E(a)

Source: Switzerland - Swiss TST 2004 p1

February 18, 2020
combinatoricsSumSubsets

Problem Statement

Let SS be the set of all n-tuples (X1,...,Xn)(X_1,...,X_n) of subsets of the set {1,2,..,1000}\{1,2,..,1000\}, not necessarily different and not necessarily nonempty. For a=(X1,...,Xn)a = (X_1,...,X_n) denote by E(a)E(a) the number of elements of X1∪...∪XnX_1\cup ... \cup X_n. Find an explicit formula for the sum ∑a∈SE(a)\sum_{a\in S} E(a)
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