HMMT 十一月 2025 · 团队赛 · 第 5 题

HMMT November 2025 — Team Round — Problem 5

[40] Kelvin the frog is in the bottom-left cell of a 6 × 6 grid, and he wants to reach the top-right cell. He can take steps either up one cell or right one cell. However, there is a raccoon in one of the 36 cells uniformly at random, and Kelvin’s path must avoid this raccoon. Compute the expected number of distinct paths Kelvin can take to reach the top-right cell. (If the raccoon is in either the bottom-left or top-right cell, then there are 0 such paths.) ◦

解析

[40] Kelvin the frog is in the bottom-left cell of a 6 × 6 grid, and he wants to reach the top-right cell. He can take steps either up one cell or right one cell. However, there is a raccoon in one of the 36 cells uniformly at random, and Kelvin’s path must avoid this raccoon. Compute the expected number of distinct paths Kelvin can take to reach the top-right cell. (If the raccoon is in either the bottom-left or top-right cell, then there are 0 such paths.) Proposed by: Derek Liu Answer: 175 10 Solution: There are = 252 paths if we disregard the raccoon. Each path traverses 11 cells, so 5 25 there is a chance the path avoids the raccoon. By linearity of expectation, the expected number of 36 such paths is 25 · 252 = 175 . 36 ◦