HMMT 十一月 2011 · 冲刺赛 · 第 28 题
HMMT November 2011 — Guts Round — Problem 28
题目详情
英文原题
[14] Determine the value of 2011/summationdisplay k=1k−1 k!(2011 −k)!.
解析
英文解析
- [ 14 ] Determine the value of
2011
∑
k − 1
.
k !(2011 − k )!
k =1
2010
2009 2 +1
( )
Answer: We note that
2011!
2011 2011
∑ ∑
k − 1 (2011!)( k − 1)
(2011!) =
k !(2011 − k )! k !(2011 − k )!
k =1 k =1
2011 2011
∑ ∑
k (2011)! 2011!
= −
k !(2011 − k )! k !(2011 − k )!
k =1 k =1
( ) ( )
2011 2011
∑ ∑
2011 2011 = k −
k kk =1 k =1
2010 2011 = (2011)(2 ) − (2 − 1)
( ( ) )
2010
Thus, we get an answer of 2009 2 + 1 / (2011!).
Note: To compute the last two sums, observe that
( )
2011
∑
2011
2011 2011 = (1 + 1) = 2
k =0 kby the Binomial Theorem, and that
( )
( ) ( ) ( )
2011 2011 2011
∑ ∑ ∑
( )
2011 1 2011 2011
2010
k = k + (2011 − k ) = 2011 2 .
k 2 k 2011 − kk =0 k =0 k =0