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 5 of 5: Substitute the required ground-set size
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s(n,1998)=1998(21998n−21997n)s(n,1998)=1998\left(2^{1998n}-2^{1997n}\right)
Detailed analysis

Putting m=1998m=1998 gives the requested sum 1998(21998n−21997n)1998(2^{1998n}-2^{1997n}).