A bag contains 2 Apples, 3 Oranges and 4 Bananas. The number of ways in which 3 fruits can be selected if atleast one banana is always in the combination (Assume fruits of same species to be alike) is

Options

A. 6
B. 10
C. 29
D. 7

Answer

To solve the problem of selecting 3 fruits from a bag containing 2 apples, 3 oranges, and 4 bananas, with the condition that at least one banana must be included, we can break down the solution into manageable steps.

### Step-by-Step Solution:

1. **Understand the Problem**: We have 2 apples, 3 oranges, and 4 bananas. We need to select 3 fruits with at least one banana included.

2. **Set Up Cases**: Since we need at least one banana in our selection, we can break this down into two cases based on the number of bananas selected:
- **Case 1**: Selecting 1 banana and 2 other fruits (which can be apples or oranges).
- **Case 2**: Selecting 2 bananas and 1 other fruit (which can be either an apple or an orange).

3. **Case 1 - One Banana**:
- We select 1 banana and need to choose 2 more fruits from the remaining apples and oranges.
- The possible combinations are:
- **Both fruits are apples**: This is not possible since we only have 2 apples, and we need to select 2. Therefore, we can select 2 apples in only 1 way (2C2 = 1).
- **Both fruits are oranges**: This is also not possible since we only have 3 oranges, and we need to select 2. Therefore, we can select 2 oranges in only 1 way (3C2 = 3).
- **One apple and one orange**: We can select 1 apple and 1 orange in 1 way (2C1 * 3C1 = 2 * 3 = 6).

- Total for Case 1:
- Selecting 1 banana + 2 apples: 1 way
- Selecting 1 banana + 2 oranges: 3 ways
- Selecting 1 banana + 1 apple + 1 orange: 6 ways
- Total = 1 + 3 + 6 = 10 ways.

4. **Case 2 - Two Bananas**:
- We select 2 bananas and need to choose 1 more fruit from the apples or oranges.
- The possible combinations are:
- **1 apple**: We can select 1 apple in 2 ways (2C1 = 2).
- **1 orange**: We can select 1 orange in 3 ways (3C1 = 3).

- Total for Case 2:
- Selecting 2 bananas + 1 apple: 2 ways
- Selecting 2 bananas + 1 orange: 3 ways
- Total = 2 + 3 = 5 ways.

5. **Final Count**:
- Total ways from Case 1 (10) + Total ways from Case 2 (5) = 10 + 5 = 15 ways.

### Conclusion:
The total number of ways to select 3 fruits with at least one banana is **15**.