Question

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 rook. You may leave unevaluated binomial coefficients in your answer.

Answer 1

Answer: $2*8^{8}-8!$

Step-by-step explanation:

We can place a rook in each row in $8^{8}$ ways.

We can place a rook in each column in $8^{8}$ ways.

So the total ways in which we can place a rook in a row or column is $2*8^{8}$

Now there are 8 ways to choose the column for the first row, 7 ways to choose the column for the second row, and so on. So there are 8! ways to choose a column for a row.

So, we get the final answer by subtracting the 8! from the total ways a rook can be placed which is $2*8^{8}-8!$ when binomial coefficients are not evaluated