In how many ways is it possible to choose a white square and a black square on a chess board so that the squares must not lie in the same row or column?

Options

A. A) 56
B. B) 896
C. C) 60
D. D) 768

Answer

To solve the problem of choosing a white square and a black square on a chessboard such that they do not lie in the same row or column, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Squares**:
A standard chessboard has 64 squares, consisting of 32 white squares and 32 black squares.

2. **Choose a White Square**:
We can choose any one of the 32 white squares. Therefore, the number of ways to choose a white square is:
\[ \text{Ways to choose a white square} = 32 \]

3. **Determine Restrictions for Choosing a Black Square**:
Once a white square is chosen, we must ensure that the black square is not in the same row or column as the chosen white square.

4. **Count the Number of Black Squares Available**:
Each row of the chessboard contains 8 squares, and since there are 8 rows, if we have chosen a white square, it occupies one square in its row and one square in its column. This means:
- 1 square in the same row is blocked (the row of the chosen white square).
- 1 square in the same column is blocked (the column of the chosen white square).

Since the white square is white, there are 8 squares in that row, and 8 squares in that column, but one of those squares is the white square itself. Therefore, the number of black squares that can still be chosen is:
\[ \text{Available black squares} = 32 - 8 = 24 \]

5. **Calculate the Total Number of Combinations**:
The total number of ways to choose a white square and a black square, ensuring they are not in the same row or column, is the product of the number of ways to choose each square:
\[ \text{Total ways} = \text{Ways to choose a white square} \times \text{Available black squares} = 32 \times 24 = 768 \]

### Final Answer:
Thus, the total number of ways to choose a white square and a black square on a chessboard such that they do not lie in the same row or column is **768**.

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