Counting Ways to Place 8 Rooks on a Chess Board without Attacks

[cragar]

Homework Statement

How many ways can you place 8 rooks on a chess board so that no 2 rooks attack each other.
Assume the rooks are identical.
Chess board is 8 by 8.

The Attempt at a Solution

To place the first rook I would have 64 choices.
for the second rook I would have 49 choices because I would be limited to a 7 by 7 square.
So the answer should be (64)(49)(36)(25)(16)(9)(4)(1).
I don't think I need to divide by 8! because when I place the rooks that is already taken care of.

 

[LCKurtz]

Homework Statement

How many ways can you place 8 rooks on a chess board so that no 2 rooks attack each other.
Assume the rooks are identical.
Chess board is 8 by 8.

The Attempt at a Solution

To place the first rook I would have 64 choices.
for the second rook I would have 49 choices because I would be limited to a 7 by 7 square.
So the answer should be (64)(49)(36)(25)(16)(9)(4)(1).
I don't think I need to divide by 8! because when I place the rooks that is already taken care of.

I don't think you need to divide by 8! either, assuming the chess board is oriented so you know which side is black and which white. That way each square is uniquely identified and I agree with your solution. However, if you are just presented with a chess board with no orientation, then you would have to account for rotations which you now couldn't distinguish.

 

[berkeman]

Homework Statement

How many ways can you place 8 rooks on a chess board so that no 2 rooks attack each other.
Assume the rooks are identical.
Chess board is 8 by 8.

The Attempt at a Solution

To place the first rook I would have 64 choices.
for the second rook I would have 49 choices because I would be limited to a 7 by 7 square.
So the answer should be (64)(49)(36)(25)(16)(9)(4)(1).
I don't think I need to divide by 8! because when I place the rooks that is already taken care of.

Interesting problem!

I get 8+8 = 16. Are you sure you can place all 8 rooks using your method?

 

[cragar]

you think the answer is 16? I mean when you place a rook down you eliminate that row and column for your next rook.

 

[berkeman]

you think the answer is 16? I mean when you place a rook down you eliminate that row and column for your next rook.

Correct. So what relationship do all the rooks have to have to one another in order not to be able to attack each other?

 

[cragar]

Each rook has to be in a different row and column, So the first rook has 8 choices for the row, then the next rook has 7 choices for the row ect... it seems like the answer might be 8!. Thanks for your help by the way, I am kinda worried about the region where the places the rooks can attack and how that over laps with rooks not in their row or column

 

[berkeman]

Each rook has to be in a different row and column, So the first rook has 8 choices for the row, then the next rook has 7 choices for the row ect... it seems like the answer might be 8!. Thanks for your help by the way, I am kinda worried about the region where the places the rooks can attack and how that over laps with rooks not in their row or column

I still get 8+8 = 16. Think geometrically...

 

[LCKurtz]

Each rook has to be in a different row and column, So the first rook has 8 choices for the row, then the next rook has 7 choices for the row ect... it seems like the answer might be 8!. Thanks for your help by the way, I am kinda worried about the region where the places the rooks can attack and how that over laps with rooks not in their row or column

It turns out that a lot of study has gone into this problem. Your thinking here for 8! is correct for an oriented chessboard (which is equivalent to dividing your original calculation by 8!). Google rooks on a chessboard for more information than you will want.

 

[berkeman]

After a PM with LCKurtz, I see now that there are more than the 16 positions that I was thinking of. Thanks!