HMMT 二月 2007 · COMB 赛 · 第 9 题

英文解析

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
   ( )2
   Answer: 11628 = . We view choosing five objects from a row of 19 objects in an unusual way.19
   First, remove two of the chosen objects, the second and fourth, which are not adjacent nor at either 5
   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,
   ( )
   the answer is = 11628.19
   ∑5
   A simple generating functions argument is also possible. Let s = ijk . Thenni + j + k = n
    
   ( )3
   ∑ ∑33
   x xn n
    
   s x = nx = = ,
   2 6 n
   (1 − x ) (1 − x )
   n ≥ 0 n ≥ 0
   ( ( ) ) ( )
   ( )
   6 n + 2
   and so s = = , yielding s = .19
   n 17
   n − 3 55