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PUMaC 2019 · 组合（B 组） · 第 2 题

PUMaC 2019 — Combinatorics (Division B) — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. Suppose Alan, Michael, Kevin, Igor, and Big Rahul are in a running race. It is given that exactly one pair of people tie (for example, two people both get second place), so that no other pair of people end in the same position. Each competitor has equal skill; this means that each outcome of the race, given that exactly two people tie, is equally likely. The probability that Big Rahul gets first place (either by himself or he ties for first) can be expressed in the form m/n , where m, n are relatively prime, positive integers. Compute m + n .
解析
  1. Suppose Alan, Michael, Kevin, Igor, and Big Rahul are in a running race. It is given that exactly one pair of people tie (for example, two people both get second place), so that no other pair of people end in the same position. Each competitor has equal skill; this means that each outcome of the race, given that exactly two people tie, is equally likely. The probability that Big Rahul gets first place (either by himself or he ties for first) can be expressed in the form m/n , where m, n are relatively prime, positive integers. Compute m + n . Proposed by Alan Chung. Answer: 5 . Solution: We compute the total number of ways the contestants to finish the race. There are ( ) 5 ways to choose the two that tie. Then treating the two people who tie as one unit, there 2 are 4! ways to arrange the 3 people and the tie in a row, which is 240 ways. There are two cases for Big Rahul to finish first; either he ties for first or he wins first place alone. If he ties for first, there are 4 people he can tie with, and then there are 3! ways to arrange the last 3 people. Thus there are 4 · 3! = 24 ways for this case. ( ) 4 If he finishes first alone, then there are ways to choose the tie, and then 3! way to arrange 2 ( ) 4 the two people who don’t tie and the pair that ties. Thus there are · 3! = 36 ways for this 2 case. Thus there are 60 total ways for Big Rahul to finish first, for a probability of 1/4, so the answer is 5.