HMMT 二月 2022 · 冲刺赛 · 第 3 题

HMMT February 2022 — Guts Round — Problem 3

[5] Herbert rolls 6 fair standard dice and computes the product of all of his rolls. If the probability that a the product is prime can be expressed as for relatively prime positive integers a and b , compute 100 a + b . b

解析

[5] Herbert rolls 6 fair standard dice and computes the product of all of his rolls. If the probability a that the product is prime can be expressed as for relatively prime positive integers a and b , compute b 100 a + b . Proposed by: Akash Das Answer: 2692 Solution: The only way this can happen is if 5 of the dice roll 1 and the last die rolls a prime number (2, 3, or 5). There are 6 ways to choose the die that rolls the prime, and 3 ways to choose the prime. 3 · 6 1 Thus, the probability is = . 6 6 2592