HMMT 二月 2002 · 冲刺赛 · 第 57 题

HMMT February 2002 — Guts Round — Problem 57

[9] How many ways, without taking order into consideration, can 2002 be expressed as the sum of 3 positive integers (for instance, 1000 + 1000 + 2 and 1000 + 2 + 1000 are considered to be the same way)? a n − 1

解析

How many ways, without taking order into consideration, can 2002 be expressed as the sum of 3 positive integers (for instance, 1000 + 1000 + 2 and 1000 + 2 + 1000 are considered to be the same way)? Solution: Call the three numbers that sum to 2002 A , B , and C . In order to prevent redundancy, we will consider only cases where A ≤ B ≤ C . Then A can range from 1 to 3( A − 1) 667, inclusive. For odd A , there are 1000 − possible values for B . For each choice 2 of A and B , there can only be one possible C , since the three numbers must add up to a fixed value. We can add up this arithmetic progression to find that there are 167167 possible 3 A combinations of A, B, C , for odd A . For each even A , there are 1002 − possible values 2 for B . Therefore, there are 166833 possible combinations for even A . In total, this makes 334000 possibilities. a n − 1