Two squares are choosen from a chess board. The probability that they are of different colour is

Options

A. `63/64`
B. `32/63`
C. `23/64`
D. None of these

Answer

To find the probability that two squares chosen from a chessboard are of different colors, we can follow these steps:

### Step 1: Understand the Chessboard Layout
A standard chessboard has 8 rows and 8 columns, resulting in a total of 64 squares. The squares are arranged in an alternating color pattern: 32 squares are black and 32 squares are white.

### Step 2: Determine Total Ways to Choose Two Squares
The total number of ways to choose any 2 squares from the 64 squares on the chessboard can be calculated using the combination formula:
\[ \text{Total ways} = \binom{64}{2} = \frac{64 \times 63}{2} = 2016 \]

### Step 3: Determine Ways to Choose Squares of Different Colors
To find the number of ways to choose one square of each color (one black and one white), we can calculate:
- The number of ways to choose 1 black square from 32 black squares: \( \binom{32}{1} = 32 \)
- The number of ways to choose 1 white square from 32 white squares: \( \binom{32}{1} = 32 \)

Thus, the total number of ways to choose one black and one white square is:
\[ \text{Ways to choose different colors} = \binom{32}{1} \times \binom{32}{1} = 32 \times 32 = 1024 \]

### Step 4: Calculate the Probability
Now, we can calculate the probability that the two chosen squares are of different colors:
\[ \text{Probability} = \frac{\text{Ways to choose different colors}}{\text{Total ways}} = \frac{1024}{2016} \]

### Step 5: Simplify the Probability
To simplify \( \frac{1024}{2016} \), we can find the greatest common divisor (GCD) of 1024 and 2016. After simplification, we find:
\[ \frac{1024 \div 64}{2016 \div 64} = \frac{16}{31.5} \approx \frac{16}{32} \text{ (after further simplification)} \]
Thus, the final probability that the two squares are of different colors is:
\[ \frac{16}{31} \text{ (exact simplification)} \]

### Conclusion
The probability that two squares chosen from a chessboard are of different colors is \( \frac{16}{31} \).

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