sheldon ross, chapter 2, question 17

Question

If 8 rooks are randomly placed on a chessboard, compute the probability that none of the rooks can caputre any of the others. That is compute the probability that no row or file contains more than one rook.

Comments

you can check this QS comment section : https://gateoverflow.in/128051/placing-rooks-in-nxn-chess-board

and answer for above QS is fairly straightforward

Answer 1

Since rooks attack horizontally & vertically, you can't have a rook in the same row or column as another.

So in the first row, you place a rook. There are 8 possible places for first one. In the next row you place a rook,it can't be in the same column as other rook so for this one we have 7 possibilities (7 places). Keep doing like this we simply have 8*7*6*5*4*3*2*1 i.e. 8! possible ways.

To place the rooks safely so that none of them clashes (capture any of the others) we have 8! ways.

Now total number of possible ways to place all 8 rooks  is simply 64 choose 8.

so the probability that no rooks capturing= 8!/64C8

---->>9.109X10^-6