2024 SSMO Team Round Problems/Problem 13

Problem

In a deck of 54 cards (2 identical jokers, 4 identical cards with $1,2,3,\dots,13$), each card is dealt to one of 3 people, each having a $\frac{1}{3}$ chance of receiving each card. If the expected sum of the number of unique cards the three of them have can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution