HMMT 二月 2020 · 冲刺赛 · 第 5 题

HMMT February 2020 — Guts Round — Problem 5

[5] Alice, Bob, and Charlie roll a 4, 5, and 6-sided die, respectively. What is the probability that a number comes up exactly twice out of the three rolls? ◦

解析

[5] Alice, Bob, and Charlie roll a 4, 5, and 6-sided die, respectively. What is the probability that a number comes up exactly twice out of the three rolls? Proposed by: Andrew Lin 13 Answer: 30 Solution 1: There are 4 · 5 · 6 = 120 different ways that the dice can come up. The common number can be any of 1, 2, 3, 4, or 5: there are 3 + 4 + 5 = 12 ways for it to be each of 1, 2, 3, or 4, because we pick one of the three people’s rolls to disagree, and there are 3, 4, and 5 ways that roll can come up (for Alice, Bob, and Charlie respectively). Finally, there are 4 ways for Bob and Charlie to both roll a 5 and Alice to roll any number. Thus there are 52 different ways to satisfy the problem condition, 52 13 and our answer is = . 120 30 Solution 2: If Bob rolls the same as Alice, Charlie must roll a different number. Otherwise Charlie must roll the same as either Alice or Bob. So the answer is 1 5 4 2 13 · + · = . 5 6 5 6 30 Solution 3: By complementary counting, the answer is 1 1 4 4 13 1 − · − · = . 5 6 5 6 30 The first and second products correspond to rolling the same number three times and rolling three distinct numbers, respectively. ◦