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 for relatively prime positive integers and find

Solution

Note that there are possible places to seat. After Andre is assigned a seat at random, there are 39 remaining seats, only one of which is in the same chair as Emily. Thus, our answer is ~pinkpig