2023 WSMO Speed Round Problems/Problem 2

Problem

There are 4 tables and 5 chairs at each table. Each chair seats 2 people. There are 10 people who are seated randomly. Andre and Emily are 2 of them, and are a couple. If the probability that Andre and Emily are in the same chair is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution