2025 SSMO Speed Round Problems/Problem 8

Problem

Let $S$ be the set of all ordered pairs $(P,Q),$ where $P$ and $Q$ are subsets of $\{1,2,\dots, 25\}$ satisfying $|P\cup Q| = 17$ and $|(P\cap Q)\cap \{20,25\}|\ge 1$ . If an ordered pair $(A,B)$ is chosen randomly from $S,$ the expected value of $|A\cap B|$ is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

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2025 SSMO Speed Round Problems/Problem 8

Problem

Solution