2025 SSMO Team Round Problems/Problem 13

Problem

Let $S$ be the set of all ordered quintuples of integers $(x_1,x_2,x_3,x_4,x_5)$ satisfying $0 \le x_1\le x_2\le x_3\le x_4 \le x_5\le 5$ . The conjugate of a quintuple in $S$ is defined as $(y_1,y_2,y_3,y_4,y_5),$ where for each integer $1\le i \le 5,$ $y_i$ is the number of indices $1\le j\le 5$ satisfying $x_j\ge i$ . A randomly chosen quintuple of $S$ is a permutation of its conjugate with probability $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

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2025 SSMO Team Round Problems/Problem 13

Problem

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