In how how many ways one black and one white rook can be placed on a chessboard so that they are never in an attacking position?

Options

A. 1234
B. 3136
C. 9516
D. 1024

Answer

To solve the problem of how many ways one black rook and one white rook can be placed on a chessboard such that they are never in an attacking position, we can follow these steps:

### Step 1: Understand the Chessboard Layout
A standard chessboard consists of 8 rows and 8 columns, making a total of 64 squares.

**Hint:** Remember that a rook attacks all squares in its row and column.

### Step 2: Place the First Rook
We can place the first rook (let's say the white rook) on any of the 64 squares of the chessboard.

**Hint:** Consider how many options you have for placing the first rook.

### Step 3: Determine Attacking Positions
Once the white rook is placed, it attacks all squares in the same row and column. Since there are 8 squares in a row and 8 squares in a column, and the square where the white rook is placed is counted in both, the total number of attacking positions is:
- 7 squares in the same row (excluding the square occupied by the white rook)
- 7 squares in the same column (excluding the square occupied by the white rook)

Thus, the total number of attacking positions for the white rook is:
\[ 7 + 7 = 14 \]

**Hint:** Make sure to account for the square occupied by the first rook when counting attacking positions.

### Step 4: Calculate Non-Attacking Positions for the Second Rook
After placing the white rook, the remaining squares where the black rook can be placed are:
\[ 64 - 1 - 14 = 49 \]
This is because we subtract the 1 square occupied by the white rook and the 14 squares that are attacked by it.

**Hint:** Think about how many squares are left after considering the attacking positions.

### Step 5: Calculate Total Arrangements
Now, since we can place the white rook in any of the 64 squares, and for each placement of the white rook, there are 49 valid squares for the black rook, the total number of ways to place both rooks is:
\[ 64 \times 49 = 3136 \]

**Hint:** Combine the choices for both rooks to find the total arrangements.

### Final Answer
Thus, the total number of ways to place one black rook and one white rook on a chessboard such that they are never in an attacking position is **3136**.