There are 5 red and 6 black balls in a bag. In how many ways 6 balls can be selected if there are at least 2 balls of each colour?

Answer

To solve the problem of selecting 6 balls from a bag containing 5 red and 6 black balls, with the condition that at least 2 balls of each color must be included, we can break down the solution into a few steps.

### Step-by-Step Solution:

1. **Understand the Requirements**: We need to select a total of 6 balls, ensuring that there are at least 2 red balls and at least 2 black balls.

2. **Determine Possible Cases**: Since we need at least 2 balls of each color, we can have the following combinations:
- Case 1: 2 red balls and 4 black balls
- Case 2: 3 red balls and 3 black balls
- Case 3: 4 red balls and 2 black balls

3. **Calculate Combinations for Each Case**:
- **Case 1**: Selecting 2 red balls and 4 black balls:
- The number of ways to choose 2 red balls from 5: \( \binom{5}{2} \)
- The number of ways to choose 4 black balls from 6: \( \binom{6}{4} \)
- Total ways for Case 1: \( \binom{5}{2} \times \binom{6}{4} \)

- **Case 2**: Selecting 3 red balls and 3 black balls:
- The number of ways to choose 3 red balls from 5: \( \binom{5}{3} \)
- The number of ways to choose 3 black balls from 6: \( \binom{6}{3} \)
- Total ways for Case 2: \( \binom{5}{3} \times \binom{6}{3} \)

- **Case 3**: Selecting 4 red balls and 2 black balls:
- The number of ways to choose 4 red balls from 5: \( \binom{5}{4} \)
- The number of ways to choose 2 black balls from 6: \( \binom{6}{2} \)
- Total ways for Case 3: \( \binom{5}{4} \times \binom{6}{2} \)

4. **Calculate Each Case**:
- For Case 1:
\[ \binom{5}{2} = 10 \quad \text{and} \quad \binom{6}{4} = 15 \]
\[ \text{Total for Case 1} = 10 \times 15 = 150 \]

- For Case 2:
\[ \binom{5}{3} = 10 \quad \text{and} \quad \binom{6}{3} = 20 \]
\[ \text{Total for Case 2} = 10 \times 20 = 200 \]

- For Case 3:
\[ \binom{5}{4} = 5 \quad \text{and} \quad \binom{6}{2} = 15 \]
\[ \text{Total for Case 3} = 5 \times 15 = 75 \]

5. **Add All Cases Together**:
\[ \text{Total Ways} = 150 + 200 + 75 = 425 \]

### Final Answer:
The total number of ways to select 6 balls with at least 2 balls of each color is **425**.