HMMT 二月 2007 · COMB 赛 · 第 9 题

HMMT February 2007 — COMB Round — Problem 9

[ 7 ] Let S denote the set of all triples ( i, j, k ) of positive integers where i + j + k = 17. Compute ∑ ijk. ( i,j,k ) ∈ S

解析

[ 7 ] Let S denote the set of all triples ( i, j, k ) of positive integers where i + j + k = 17. Compute ∑ ijk. ( i,j,k ) ∈ S 2 ( ) 19 Answer: 11628 = . We view choosing five objects from a row of 19 objects in an unusual way. 5 First, remove two of the chosen objects, the second and fourth, which are not adjacent nor at either end, forming three nonempty groups of consecutive objects. We then have i , j , and k choices for the first, third, and fifth objects. Because this is a reversible process taking a triple ( i, j, k ) to ijk choices, ( ) 19 the answer is = 11628. 5 ∑ A simple generating functions argument is also possible. Let s = ijk . Then n i + j + k = n   3 ( ) 3 3 ∑ ∑ x x n n   s x = nx = = , n 2 6 (1 − x ) (1 − x ) n ≥ 0 n ≥ 0 ( ( ) ) ( ) ( ) 6 n + 2 19 and so s = = , yielding s = . n 17 5 n − 3 5