Hey I have this question that I don\'t quite understand can anyone please be of
ID: 3122037 • Letter: H
Question
Hey I have this question that I don't quite understand can anyone please be of assistance?
A rook is a chess piece that can attack another piece if they are in the same row or the same column. A chess board has 8 rows and 8 columns. (a) How many ways can you place 8 rooks so that no two can attack each other? We assume that the rooks are identical so interchanging two rooks does not count as a separate placement. (b) Suppose that you have 4 wooden and 4 marble rooks. How many ways can you do this? (c) How many ways can you do this if all the 8 rooks are different?Explanation / Answer
(a) If a rook is placed in one row, that row is occupied. Same applies for a column.
So if the number of ways in which 1 rook can be placed are 8. And once a rook is placed, the chess board "shinks" by a row and a column i.e becomes a 7x7 board for the rest of the rooks. This process continues
=> Total number of ways = 8*7*6*....1 = 8!
(c) The solution for this problem is same as (a) repeated 8 times for every column.
= 8! * 8
(b) The solution for b is half times the solution for (c) since there are 2 groupings
= 8! * 4