Rooks Attacking Every Square on a Chess Board 8 rooks are randomly placed on different squares of a chessboard. A rook is said to attack all of the squares in its row and its column.  Compute the probability that every square is occupied or attacked by at least 1 rook.
The first step I took was to state that there are $64C8$ ways to decide how to place the 8 rooks on the chessboard. Next, I tried to experiment with a physical chessboard to see how this could be done. The only way I found that every square on the board can be attacked is if one rook is in either every horizontal row or every vertical row. Therefore, there are $2 * 8^8 - 2$ ways to place the rooks. To clarify, it is "-2" because the diagonals are counted twice. 
Is there a case that I overlooked, or did I solve the problem correctly?
Thanks,
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Consider e.g. In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other? for the number of ways that 8 rooks can be placed so that all squares are attacked or occupied.
Then, how many ways in total can you place 8 rooks regardless of attacked or occupied squares?
