In how many ways is it possible to make a selection by taking any number of 15 fruits, namely 3 oranges, 5 apples and 7 mangoes?

Options

A. 131
B. 191
C. 68
D. 3720

Answer

To solve the problem of how many ways it is possible to make a selection by taking any number of 15 fruits (3 oranges, 5 apples, and 7 mangoes), we can use the concept of combinations with repetition.

### Step-by-Step Solution:

1. **Identify the Types of Fruits and Their Quantities**:
- Oranges: 3
- Apples: 5
- Mangoes: 7

2. **Understanding the Selection**:
- We can select from 0 to 3 oranges, 0 to 5 apples, and 0 to 7 mangoes.
- This means we can choose any number of each type of fruit, including not choosing any at all.

3. **Calculate the Number of Choices for Each Fruit**:
- For oranges, since we can choose from 0 to 3, we have 4 options (0, 1, 2, or 3).
- For apples, since we can choose from 0 to 5, we have 6 options (0, 1, 2, 3, 4, or 5).
- For mangoes, since we can choose from 0 to 7, we have 8 options (0, 1, 2, 3, 4, 5, 6, or 7).

4. **Total Combinations**:
- To find the total number of ways to select the fruits, we multiply the number of options for each type of fruit:
\[ \text{Total selections} = (\text{Choices for oranges}) \times (\text{Choices for apples}) \times (\text{Choices for mangoes}) \]
\[ = 4 \times 6 \times 8 \]

5. **Calculate the Result**:
- Performing the multiplication:
\[ 4 \times 6 = 24 \]
\[ 24 \times 8 = 192 \]

6. **Final Answer**:
- Therefore, the total number of ways to make a selection of the fruits is **192**.