There are 4 white, 3 black and 3 red balls in a bag. Find the number of ways of selecting three balls, if at least one black ball is there.

Answer

To solve the problem of selecting three balls from a bag containing 4 white, 3 black, and 3 red balls with the condition that at least one black ball is included, we can break it down into cases based on the number of black balls selected.

### Step-by-Step Solution:

1. **Identify the Total Balls**:
We have:
- 4 white balls
- 3 black balls
- 3 red balls

2. **Define Cases Based on Black Balls**:
We will consider three cases based on the number of black balls selected:
- Case 1: 1 black ball
- Case 2: 2 black balls
- Case 3: 3 black balls

3. **Case 1: Selecting 1 Black Ball**:
- Choose 1 black ball from 3 black balls: \( \binom{3}{1} = 3 \)
- We need to select 2 more balls from the remaining 7 balls (4 white + 3 red).
- The combinations for selecting 2 balls can be:
- 2 white: \( \binom{4}{2} = 6 \)
- 2 red: \( \binom{3}{2} = 3 \)
- 1 white and 1 red: \( \binom{4}{1} \times \binom{3}{1} = 4 \times 3 = 12 \)
- Total for Case 1:
\[ 3 \times (6 + 3 + 12) = 3 \times 21 = 63 \]

4. **Case 2: Selecting 2 Black Balls**:
- Choose 2 black balls from 3 black balls: \( \binom{3}{2} = 3 \)
- We need to select 1 more ball from the remaining 7 balls (4 white + 3 red).
- The combinations for selecting 1 ball can be:
- 1 white: \( \binom{4}{1} = 4 \)
- 1 red: \( \binom{3}{1} = 3 \)
- Total for Case 2:
\[ 3 \times (4 + 3) = 3 \times 7 = 21 \]

5. **Case 3: Selecting 3 Black Balls**:
- Choose all 3 black balls: \( \binom{3}{3} = 1 \)
- No more balls need to be selected since we already have 3 balls.
- Total for Case 3:
\[ 1 \]

6. **Combine All Cases**:
- Total number of ways = Case 1 + Case 2 + Case 3
\[ 63 + 21 + 1 = 85 \]

### Final Answer:
The total number of ways to select three balls with at least one black ball is **85**.