HMMT 十一月 2009 · GEN1 赛 · 第 7 题

HMMT November 2009 — GEN1 Round — Problem 7

[ 6 ] There are 15 stones placed in a line. In how many ways can you mark 5 of these stones so that there are an odd number of stones between any two of the stones you marked?

解析

[ 6 ] There are 15 stones placed in a line. In how many ways can you mark 5 of these stones so that there are an odd number of stones between any two of the stones you marked? Answer: 77 Number the stones 1 through 15 in order. We note that the condition is equivalent ( ) 8 to stipulating that the stones have either all odd numbers or all even numbers. There are ways to 5 ( ) 7 choose 5 odd-numbered stones, and ways to choose all even-numbered stones, so the total number 5 ( ) ( ) ( ) 8 7 n of ways to pick the stones is + = 77. ( is the number of ways to choose k out of n items. It 5 5 k n ! equals ). k !( n − k )!