HMMT 二月 2007 · TEAM2 赛 · 第 4 题

HMMT February 2007 — TEAM2 Round — Problem 4

[ 20 ] Thomas and Michael are just two people in a large pool of well qualified candidates for appointment to a problem writing committee for a prestigious college math contest. It is 40 times more likely that both will serve if the size of the committee is increased from its traditional 3 members to a whopping n members. Determine n. (Each person in the pool is equally likely to be chosen.) 2 2 2

解析

[ 20 ] Thomas and Michael are just two people in a large pool of well qualified candidates for appointment to a problem writing committee for a prestigious college math contest. It is 40 times more likely that both will serve if the size of the committee is increased from its traditional 3 members to a whopping n members. Determine n. (Each person in the pool is equally likely to be chosen.) Answer: 16 . Suppose there are k candidates. Then the probability that both serve on a 3 membered ( ) ( ) ( ) k k − 2 k committee is ( k − 2) / , and the odds that both serve on an n membered committee are / . 3 n − 2 n The ratio of the latter to the former is ( )( ) k k − 2 k !( k − 2)!1!( k − 3)! n !( k − n )! n · ( n − 1) 3 n − 2 ( ) = = . k k !( k − 2)!( n − 2)!( k − n )!3!( k − 3)! 3! ( k − 2) n Solving n · ( n − 1) = 240 produces n = 16 , − 15 , and we discard the latter. 2 2 2