A bag contains 3 red, 4 white and 5 blue balls. If two balls are drawn at random, then the probability that they are of different colors, is

Options

A. `47/66`
B. `23/33`
C. `47/132`
D. `47/33`

Answer

To solve the problem of finding the probability that two balls drawn from a bag containing 3 red, 4 white, and 5 blue balls are of different colors, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Balls**:
- The bag contains:
- Red balls = 3
- White balls = 4
- Blue balls = 5
- Total number of balls = 3 + 4 + 5 = 12.

2. **Calculate Total Outcomes**:
- The total ways to draw 2 balls from 12 is given by the combination formula:
\[ \text{Total Outcomes} = \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66. \]

3. **Calculate Favorable Outcomes for Different Colors**:
- We need to consider the cases where the two balls drawn are of different colors. The possible combinations are:
- Case 1: 1 Red and 1 White
- Case 2: 1 White and 1 Blue
- Case 3: 1 Blue and 1 Red

- **Case 1: 1 Red and 1 White**:
\[ \text{Ways} = \binom{3}{1} \times \binom{4}{1} = 3 \times 4 = 12. \]

- **Case 2: 1 White and 1 Blue**:
\[ \text{Ways} = \binom{4}{1} \times \binom{5}{1} = 4 \times 5 = 20. \]

- **Case 3: 1 Blue and 1 Red**:
\[ \text{Ways} = \binom{5}{1} \times \binom{3}{1} = 5 \times 3 = 15. \]

4. **Total Favorable Outcomes**:
- Add the favorable outcomes from all cases:
\[ \text{Total Favorable Outcomes} = 12 + 20 + 15 = 47. \]

5. **Calculate the Probability**:
- The probability that the two balls drawn are of different colors is given by:
\[ P(\text{Different Colors}) = \frac{\text{Total Favorable Outcomes}}{\text{Total Outcomes}} = \frac{47}{66}. \]

### Final Answer:
The probability that the two balls drawn are of different colors is \(\frac{47}{66}\).

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