Rook Probability!
Eight rooks are placed randomly on different squares of a chessboard. If the probability that none of the rooks is under attack by another rook can be expressed as
for positive coprime integers
, then find the value of
.
Note: You may use a Calculator at the last step of your solution.
The answer is 61475079.
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1 solution
For rooks to be non-attacking, there must be a rook in every row and every column. Starting with the bottom row, the first rook can be put on any one of eight different squares. Wherever it is placed, there is the option of seven squares for the second rook in the second row. Then there are six squares from which to select the third row, and so on. Therefore the number of different ways must be 8 ! = 4 0 , 3 2 0
The total number of ways to place eight rooks is 6 4 C 8 = 4 4 2 6 1 6 5 3 6 8 .