In now many ways is it possible to make a selection by taking any number of all 20 fruits, namely 9 mangoes 7,oranges and 4 apples?

Options

A. 199
B. `""^(20)C_3`
C. `""^(20)P_3`
D. `399`

Answer

To solve the problem of how many ways it is possible to make a selection by taking any number of the 20 fruits (9 mangoes, 7 oranges, and 4 apples), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Number of Each Fruit**:
- We have 9 mangoes, 7 oranges, and 4 apples.

2. **Understanding the Selection Process**:
- For each type of fruit, we can select from 0 up to the maximum number available. This means we can choose:
- 0 to 9 mangoes (10 options: 0, 1, 2, ..., 9)
- 0 to 7 oranges (8 options: 0, 1, 2, ..., 7)
- 0 to 4 apples (5 options: 0, 1, 2, 3, 4)

3. **Calculating the Total Combinations**:
- For mangoes, the number of ways to select is 10 (0 to 9).
- For oranges, the number of ways to select is 8 (0 to 7).
- For apples, the number of ways to select is 5 (0 to 4).
- Since the selections are independent, we multiply the number of ways to select each type of fruit:
\[ \text{Total selections} = (10) \times (8) \times (5) \]

4. **Performing the Multiplication**:
- Calculate \(10 \times 8 = 80\)
- Then calculate \(80 \times 5 = 400\)

5. **Adjusting for the Empty Selection**:
- The total calculated (400) includes the case where no fruit is selected at all (0 mangoes, 0 oranges, and 0 apples). Since we want to count only the selections where at least one fruit is chosen, we subtract 1:
\[ \text{Final count} = 400 - 1 = 399 \]

### Final Answer:
The total number of ways to make a selection by taking any number of the 20 fruits is **399**.

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