HMMT-Nov 2021 gen, Problem 6

Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of 18 cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario holds at the end (including the Jokers) is $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Solution: Solution: Considering ordering the nine pairs by the time they are first complete. Since the pairs are treated equally by the drawing process, this ordering is a uniform ordering. Therefore the problem becomes the following: consider ordering $7 \mathrm{~N}$ 's and $2 \mathrm{~J}$ 's randomly. What is the expected position of the first J? We may solve this by linearity of expectation. Every $\mathrm{N}$ has exactly a $1 / 3$ chance of being in front of the $2 \mathrm{~J}$ 's, so the expected number of $\mathrm{N}$ 's before the first $\mathrm{J}$ is $7 / 3$. Thus the expected position of the first $\mathrm{J}$ is $7 / 3+1=10 / 3$. $\fbox{1003}$.

Answer: 1003