HMMT 二月 2006 · COMB 赛 · 第 2 题

英文解析

2. 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 definitionk k 0 60
   ( ) ( )
   63 63 = = 1953.
   61 2