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Subset Ordered Pairs of {1, 2, ..., 10}

Source: Putnam 1990 A6

July 12, 2013
Putnamcollege contests

Problem Statement

If XX is a finite set, let XX denote the number of elements in XX. Call an ordered pair (S,T)(S,T) of subsets of {1,2,⋯ ,n} \{ 1, 2, \cdots, n \} \emph {admissible} if s>∣T∣ s > |T| for each s∈S s \in S , and t>∣S∣ t > |S| for each t∈T t \in T . How many admissible ordered pairs of subsets {1,2,⋯ ,10} \{ 1, 2, \cdots, 10 \} are there? Prove your answer.
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