HMMT 二月 2005 · GEN1 赛 · 第 5 题

英文解析

5. In how many ways can 4 purple balls and 4 green balls be placed into a 4 × 4 grid suchthat every row and column contains one purple ball and one green ball? Only one ballmay be placed in each box, and rotations and reflections of a single configuration areconsidered different.
   Solution: 216
   There are 4! = 24 ways to place the four purple balls into the grid. Choose any purpleball, and place two green balls, one in its row and the other in its column. Thereare four boxes that do not yet lie in the same row or column as a green ball, and atleast one of these contains a purple ball (otherwise the two rows containing green ballswould contain the original purple ball as well as the two in the columns not containinggreen balls). It is then easy to see that there is a unique way to place the remaininggreen balls. Therefore, there are a total of 24 · 9 = 216 ways.