PUMaC 2021 · 组合（B 组） · 第 4 题

PUMaC 2021 — Combinatorics (Division B) — Problem 4

Eighteen people are standing in a (socially-distanced) line to enter a grocery store. Five people are wearing a black mask, 6 are wearing a gray mask, and 7 are wearing a white mask. Suppose that these 18 people got on line in a random order. The expected number of pairs of adjacent a people wearing different-colored masks can be given by , where gcd( a, b ) = 1. Compute a + b . b

解析

Eighteen people are standing in a (socially-distanced) line to enter a grocery store. Five people are wearing a black mask, 6 are wearing a gray mask, and 7 are wearing a white mask. Suppose that these 18 people got on line in a random order. The expected number of pairs of adjacent a people wearing different-colored masks can be given by . Compute a + b . b Proposed by: Nancy Xu Answer: 116 For each pair of adjacent people, the probability that they are wearing different-colored masks 5 13 6 12 7 11 107 is · + · + · = . There are 17 pairs of adjacent people, so the expected value 18 17 18 17 18 17 153 107 107 is · 17 = , and our answer is 107 + 9 = 116. 153 9