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 4 of 5: Solve the recurrence
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s(n,m)=m(2nm−2n(m−1))s(n,m)=m\left(2^{nm}-2^{n(m-1)}\right)
Detailed analysis

Induction on mm using the recurrence and the base value gives s(n,m)=m(2nm−2n(m−1))s(n,m)=m(2^{nm}-2^{n(m-1)}).