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2025 SSMO Speed Round Problems/Problem 8

Revision as of 11:10, 9 September 2025 by Pinkpig (talk | contribs) (Created page with "==Problem== Let <math>S</math> be the set of all ordered pairs <math>(P,Q),</math> where <math>P</math> and <math>Q</math> are subsets of <math>\{1,2,\dots, 25\}</math> satis...")
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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 $m+n$.

Solution

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