HMMT 十一月 2016 · GEN 赛 · 第 6 题

英文解析

6. The numbers 1 , 2 · · · 11 are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is largerthan exactly one number to its right.
   Proposed by: Allen Liu
   Answer:10
   Suppose the middle number is k . Then there are k − 1 ways to pick the number smaller than k to 33
   ( )
   11 − kits left and ways to pick the 4 numbers larger than k to its right. Hence there is a total of
   ( )4
   ∑
   11 − k 7
   ( k − 1) · ways for there to be exactly one number smaller than k to its left. We calculatek =2
   this total:4
   ( ) ( )
   7 9 j
   ∑ ∑ ∑
   11 − k i
   ( k − 1) · =
   4 4
   j =4 i =4
   k =2
   ( )
   ∑9
   j + 1
   5=
   j =4
   ( ) = .11
   The only way k can be larger than exactly one number to its right is if k = 3. Then the probability of 6
   2 ·108
   ( )
   this happening is = .4
   ( )11
   633