HMMT 十一月 2012 · GEN 赛 · 第 5 题

HMMT November 2012 — GEN Round — Problem 5

[ 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 other if and only if they are in the same row or the same column.) ◦

解析

[ 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 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 in the 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 ◦