HMMT 十一月 2012 · GEN 赛 · 第 5 题
HMMT November 2012 — GEN Round — Problem 5
题目详情
英文原题
[4] How many ways are there to arrange three indistinguishable rooks on a 6 ×6 board such that no
two rooks are attacking each other? (Two rooks are attacking each ot her if and only if they are in the
same row or the same column.)
解析
英文解析
- [ 4 ] How many ways are there to arrange three indistinguishable rooks on a 6 × 6 board such that notwo rooks are attacking each other? (Two rooks are attacking each other if and only if they are in the same row or the same column.)
Answer: 2400 There are 6 × 6 = 36 possible places to place the first rook. Since it cannot be inthe same row or column as the first, the second rook has 5 × 5 = 25 possible places, and similarly, the third rook has 4 × 4 = 16 possible places. However, the rooks are indistinguishable, so there are 3! = 6
36 × 25 × 16
ways to reorder them. Therefore, the number of arrangements is = 2400.
°6