HMMT 十一月 2009 · GEN1 赛 · 第 7 题

英文解析

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?
   Answer: 77 Number the stones 1 through 15 in order. We note that the condition is equivalent
   ( )
   to stipulating that the stones have either all odd numbers or all even numbers. There are ways to 8
   ( )5
   choose 5 odd-numbered stones, and ways to choose all even-numbered stones, so the total number 7
   ( ) ( ) ( )5
   8 7 nof ways to pick the stones is + = 77. ( is the number of ways to choose k out of n items. It
   5 5 kn !
   equals ).
   k !( n − k )!