# Thread: how many ways can rooks be placed?.

1. ## how many ways can rooks be placed?.

Here is a tricky counting problem.

"How many ways can two red and four blue rooks be placed on an 8-by-8 chessboard so that no two rooks can attack one another?".

I believe that all placements of 6 non-attacking rooks on the board would be

$\displaystyle (C(8,6))^{2}\cdot 6!$, but the two different colors are tricky. Or am I making it too difficult?.

2. Originally Posted by galactus
Here is a tricky counting problem.

"How many ways can two red and four blue rooks be placed on an 8-by-8 chessboard so that no two rooks can attack one another?".

I believe that all placements of 6 non-attacking rooks on the board would be

$\displaystyle (C(8,6))^{2}\cdot 6!$, but the two different colors are tricky. Or am I making it too difficult?.
That result is for six indistinguishable rooks (I am trusting you on that, I have not checked it). So the final answer is this times the number of ways that the six can be coloured such that two are red and four are blue, which is 15 ways.

(Given any acceptable configuration of indistinguishable rooks, they can be numbered in order from the top left in the order they appear when read from left to right down the rows. Then we have the problem of assigning the colours to the numbered rooks for this configuration )

CB

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### in how many ways can two red and four blue rooks

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