Three squares of a chessboard are chosen at random. the probability that two are of one colour and one of another is :

Options

A. `67/992`
B. `16/21`
C. `31/32`
D. none of these

Answer

To solve the problem of finding the probability that when three squares of a chessboard are chosen at random, two are of one color and one is of another, we can follow these steps:

### Step 1: Determine Total Squares on the Chessboard
A standard chessboard has 8 rows and 8 columns, resulting in a total of:
\[ 64 \text{ squares} \]

### Step 2: Calculate Total Ways to Choose 3 Squares
We need to find the total number of ways to choose 3 squares from the 64 squares. This can be calculated using combinations:
\[ \text{Total ways} = \binom{64}{3} = \frac{64!}{3!(64-3)!} = \frac{64 \times 63 \times 62}{3 \times 2 \times 1} = 39711 \]

### Step 3: Determine the Number of Favorable Outcomes
In a chessboard, there are 32 black squares and 32 white squares. We want to find the number of ways to choose 2 squares of one color and 1 square of another color.

#### Case 1: Choosing 2 Black and 1 White
- The number of ways to choose 2 black squares from 32:
\[ \binom{32}{2} = \frac{32 \times 31}{2 \times 1} = 496 \]
- The number of ways to choose 1 white square from 32:
\[ \binom{32}{1} = 32 \]
- Therefore, the total ways for this case:
\[ 496 \times 32 = 15872 \]

#### Case 2: Choosing 2 White and 1 Black
- The number of ways to choose 2 white squares from 32:
\[ \binom{32}{2} = 496 \]
- The number of ways to choose 1 black square from 32:
\[ \binom{32}{1} = 32 \]
- Therefore, the total ways for this case:
\[ 496 \times 32 = 15872 \]

### Step 4: Combine Favorable Outcomes
Now, we add the favorable outcomes from both cases:
\[ \text{Total favorable outcomes} = 15872 + 15872 = 31744 \]

### Step 5: Calculate the Probability
The probability that two squares are of one color and one square is of another color is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{31744}{39711} \]

### Final Step: Simplify the Probability
This fraction can be simplified if necessary, but for the purpose of this problem, we can leave it in this form.

### Conclusion
Thus, the probability that when three squares of a chessboard are chosen at random, two are of one color and one is of another is:
\[ P = \frac{31744}{39711} \]