HMMT 二月 2006 · COMB 赛 · 第 2 题
HMMT February 2006 — COMB Round — Problem 2
题目详情
- Compute 2 n n n n 60 3 2 1 ∑ ∑ ∑ ∑ ∑ · · · 1 . n =0 n =0 n =0 n =0 n =0 60 59 2 1 0
解析
- Compute 2 n n n n 60 3 2 1 ∑ ∑ ∑ ∑ ∑ · · · 1 . n =0 n =0 n =0 n =0 n =0 60 59 2 1 0 Answer: 1953 Solution: The given sum counts the number of non-decreasing 61-tuples of integers ( n , . . . , n ) from the set { 0 , 1 , 2 } . Such 61-tuples are in one-to-one correspondence 0 60 with strictly increasing 61-tuples of integers ( m , . . . , m ) from the set { 0 , 1 , 2 , . . . , 62 } : 0 60 simply let m = n + k . But the number of such ( m , . . . , m ) is almost by definition k k 0 60 ( ) ( ) 63 63 = = 1953. 61 2