HMMT 十一月 2016 · GEN 赛 · 第 6 题

HMMT November 2016 — GEN Round — Problem 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 larger than exactly one number to its right.

解析

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 larger than exactly one number to its right. Proposed by: Allen Liu 10 Answer: 33 Suppose the middle number is k . Then there are k − 1 ways to pick the number smaller than k to ( ) 11 − k its left and ways to pick the 4 numbers larger than k to its right. Hence there is a total of 4 ( ) ∑ 7 11 − k ( k − 1) · ways for there to be exactly one number smaller than k to its left. We calculate k =2 4 this total: ( ) ( ) j 7 9 ∑ ∑ ∑ 11 − k i ( k − 1) · = 4 4 j =4 i =4 k =2 ( ) 9 ∑ j + 1 = 5 j =4 ( ) 11 = . 6 The only way k can be larger than exactly one number to its right is if k = 3. Then the probability of 8 10 2 · ( ) 4 this happening is = . 11 ( ) 33 6