HMMT-Nov 2021 guts, Problem 12

Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Solution: Solution: The probability that all three cards drawn are face cards is $\left(\frac{3}{13}\right)^{3}=\frac{27}{2197}$. In that case, the sum is 30 and therefore a multiple of 10 . Otherwise, one of the cards is not a face card, so its point value $p$ is drawn uniformly from values from 1 to 10 . The sum of the values of the other two cards uniquely determines the point value $p$ for which the sum of all three values is a multiple of 10 . Therefore, the total probability is $\frac{27}{2197}+\frac{1}{10}\left(1-\frac{27}{2197}\right)=\frac{244}{2197}$. $\fbox{26597}$.

Answer: 26597