返回题库

HMMT 十一月 2011 · 冲刺赛 · 第 28 题

HMMT November 2011 — Guts Round — Problem 28

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 14 ] Determine the value of 2011 ∑ k − 1 . k !(2011 − k )! k =1
解析
  1. [ 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 k k =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 k =0 by the Binomial Theorem, and that ( ) ( ) ( ) ( ) 2011 2011 2011 ∑ ∑ ∑ ( ) 2011 1 2011 2011 2010 k = k + (2011 − k ) = 2011 2 . k 2 k 2011 − k k =0 k =0 k =0