PUMaC 2012 · 组合（B 组） · 第 2 题

PUMaC 2012 — Combinatorics (Division B) — Problem 2

[ 3 ] If the probability that the sum of three distinct integers between 16 and 30 (inclusive) is m even can be written as , where m and n are relatively prime positive integers, find m + n . n

解析

There are 8 even numbers and 7 odd numbers from 16 to 30. For the sum of three integers to be even, either all three must be even, or two must be odd and the last must be even. There ( ) ( ) 8 7 are = 7 · 8 ways to choose the three even numbers, and 8 · = 7 · 24 ways to choose the 3 2 ( ) 15 one even and two odd integers. In total, there are = 7 · 65 ways to choose three distinct 3 7 · 8+7 · 24 32 numbers from 15 numbers. Thus, the probability that the sum is even is = , and 7 · 65 65 the answer is m + n = 97 .