PUMaC 2014 · 团队赛 · 第 5 题

PUMaC 2014 — Team Round — Problem 5

[ 5 ] How many sets of positive integers ( a, b, c ) satisfies a > b > c > 0 and a + b + c = 103? 2 x

解析

[ 5 ] How many sets of positive integers ( a, b, c ) satisfies a > b > c > 0 and a + b + c = 103? Solution: ( ) 103+2 − 3 There are = 102 × 101 / 2 = 51 × 101 ways of spliting 103 into 3 non empty piles. 2 There are 3 × 51 ways to split such that 2 of the 3 piles are equal and no pile is empty, and it is impossible for all 3 piles to be of the same size. Hence there are 51 × (101 − 3) = 51 × 98 ways of spliting 103 into distinct, non empty piles, and ordering them, we have 51 × 98 / 6 = 17 × 49 = 833 ways of doing so. 2 x