You have eight rooks in your possession A rook can either mo

You have eight rooks in your possession. A rook can either move along a row or a column, but it can make no other movement. It can move as many spots as it wants along the row or column of its choosing. You randomly place these eight rooks on the board, which is an 8 by 8 square. Find the probability that no rook can capture (run inot with a legal move) any other rook?

*please include explanation*

Solution

there are all total 8*8=64 squares. now we need to place the rooks in such a way that no rook capture the other one.

now the first rook can be placed arbitrarily. so it can be at any one spot out of 64 spots with probability 1/64.

now the next rook can not be placed in the row or column of the previous rook. for for 2nd rook there are 7*7=49 spots left with probabilty at each spot 1/49. simillarly for the third rook we have 6*6=36 squares with probabilty 1/36 and so on. at last for the last rook we have only one spot left with certain probability 1.

hence the probability that no rook capture any other rook is (1/64)*(1/49)*(1/36)*(1/25)*(1/16)*(1/9)*(1/4)*1 [answer]