MathLabs

Problem 1

Let S\mathcal S be the set of all nn-tuples (A1,A2,…,An)(A_1,A_2,\ldots,A_n) in which each AiA_i is a subset of {1,2,…,1998}\{1,2,\ldots,1998\}. For K∈SK\in\mathcal S, let f(K)=∣A1∪A2∪⋯∪An∣f(K)=|A_1\cup A_2\cup\cdots\cup A_n|. Find ∑K∈Sf(K)\sum_{K\in\mathcal S}f(K).
Step 3 of 5: Set the initial value
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s(n,1)=2n−1s(n,1)=2^n-1
Detailed analysis

For m=1m=1, the union has size 11 exactly when at least one of the nn sets contains 11. Thus s(n,1)=2n−1s(n,1)=2^n-1.