HMMT 十一月 2008 · GEN2 赛 · 第 3 题

英文解析

3. [ 5 ] Joe B. now clears the board. How many ways can he place 3 white rooks and 3 black rooks on theboard so that no two rooks of opposite color can attack each other?
   Answer: 608 Consider first placing the white rooks. They will occupy either 3 columns and 1
   row, 3 columns and 2 rows, 3 columns and 3 rows, 2 rows and 2 columns, 2 columns and 3 rows, or
   1 column and 3 rows. First note that placing the black rooks is impossible in the second, third, andfifth cases. The first case can happen in 4 · 4 ways, and each leads to a unique way to place the black
   ( )
   rooks. In the fourth case, we can choose the row with 2 rooks in 4 ways, place the rooks in ways,4
   ( )2
   choose the column of the other rook in 2 ways, and place it in 3 ways, for a total of 4 · · 2 · 3 = 1444
   ways to place the white rooks in this configuration; the black rooks can then be placed in any of 42
   possible locations, and there are 4 ways to do this, leading to 576 possibilities in this case. Finally, thesixth case is analogous to the first, giving 16 possibilities. Summing, we find 16 + 576 + 16 = 608 totalplacements.