Table of Contents

- 1 How many ways can you place n rooks on a chessboard so that no two attack each other?
- 2 How many ways can you place two identical rooks?
- 3 How many ways are there to place 5 rooks on a chessboard?
- 4 How many ways can 4 rooks be placed on a chessboard so that no two are in the same row or the same column?
- 5 How many ways can you place 5 Rooks on a chessboard?
- 6 How many ways can you place 5 rooks on a chessboard?

## How many ways can you place n rooks on a chessboard so that no two attack each other?

40320 ways

The rook polynomial as a generalization of the rooks problem Indeed, its result is that 8 non-attacking rooks can be arranged on an 8 × 8 chessboard in r8 = 8! = 40320 ways.

### How many ways can you place two identical rooks?

One of the 2 rooks can be placed on the board in 64 ways ( each square is one possible way ).

**How many ways can you place 4 mutually non-attacking rooks on the chessboard using only the outer edges of the board?**

At most 4 non-attacking rooks can be placed on a 5 × 4 board and they can be placed in 5 · 4 · 3 · 2 = 5! ways.

**How many ways are there to put one white and one black king on a chessboard so that they do not attack each other?**

So we have to divide it by 2. Hence actual possible ways will be 32*49.

## How many ways are there to place 5 rooks on a chessboard?

Then I calculated the number of ways in which 5 rooks can be placed by leaving both first row and first column empty, which then corresponds to placing 5 rooks on 7×7 board (because first row and column are out of picture), this can be done in (75)5! ways. Then I subtracted this from total, giving me 4200 ways.

### How many ways can 4 rooks be placed on a chessboard so that no two are in the same row or the same column?

It is immaterial whether it is in the center, on an edge, or even in a corner. So, for any square you choose for the rook there are 49 (64-15) squares that you can place an opposing rook without having them threaten each other.

**How many ways are there to place 2 identical rooks in a common row or column?**

There are 64 ways to place the black rook. For each such way, there are 14 ways to place the white rook, for a total of (64)(14). But the rooks are probably intended to be identical. Thus the number of black rook/white rook placements must be divided by 2, for a total of (64)(14)2.

**How many ways are there to place 2 identical rooks in a common row or column of an 8×8 chess board?**

28! =40320 ways. As you have 8 rows and 8 rooks and no two rooks can be on the same row, each row should have exactly one rook.

## How many ways can you place 5 Rooks on a chessboard?

### How many ways can you place 5 rooks on a chessboard?

**How many ways are there to place 2 identical kings on an 8 8 chessboard so that the Kings are not in adjacent squares?**

(1) How many ways are there of placing two Kings on an 8×8 chessboard so that they are not on adjacent squares? (4 · 60 + 24 · 58 + 36 · 55)/2 = 1806.

**How many ways are there to place two identical rooks on a standard checkerboard so that they are attacking each other?**

=8×7×6×5×4×3×2×1=40320. For two rooks to be attacking each other, they must either share a row or a column.