# Enumeration problem: Two Rooks on Chess Board

How many ways are there to place two rooks on the same column or same row on a 8 x 8 chess board.

• There are two possible answers, depending on whether the rooks are to be viewed as identical, or different (black and white). Probably identical is intended. – André Nicolas Sep 21 '13 at 6:37
• What does identical mean? – gandolf Sep 22 '13 at 2:52
• It means that we cannot tell the rooks apart, so only their location matters. The situation is different if we imagine the rooks to be say white and black. In that case, there are twice as many arrangements as when the rooks look the same. – André Nicolas Sep 22 '13 at 3:39
• Okay, that makes total sense, thanks for the help. – gandolf Sep 25 '13 at 8:06

First imagine that the rooks have student numbers, or that one is black and the other is white.

There are $64$ ways to place the black rook. For each such way, there are $14$ ways to place the white rook, for a total of $(64)(14)$.

But the rooks are probably intended to be identical. Thus the number of black rook/white rook placements must be divided by $2$, for a total of $\frac{(64)(14)}{2}$.

• Okay thanks for the help. I'm still want to clear some uncertainties about the whole identical rooks part of the problem. Does this mean we eliminate positions that have already been used with one rook. – gandolf Sep 21 '13 at 19:43
• The product $(64)(14)$ counts the number of ways to place a black rook and a white rook. If a person is colour-blind, she will see black on b2, white on b7 as being the same as white on b2, black on b7. So the number of "coloured" arrangements is twice the number of monochrome arrangements. Counting coloured arrangements is a little easier, so we count them and divide by $2$. – André Nicolas Sep 21 '13 at 22:12

HINT: There are $16$ ways to choose a row or a column.

• Once you’ve picked a row or column, how many ways are there to pick two squares in that row or column?

• How do you combine that number with the $16$ to get the final answer?