In how many ways can a selection be made out of 5 oranges , 8 mangoes and 7 plantains?

Answer

To solve the problem of how many ways a selection can be made out of 5 oranges, 8 mangoes, and 7 plantains, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Selection Options**:
- For each type of fruit, we can either select a certain number of fruits or choose not to select any.
- We have:
- 5 oranges
- 8 mangoes
- 7 plantains

2. **Calculate the Selection Ways for Oranges**:
- We can select from 0 to 5 oranges. This gives us:
- 0 oranges: 1 way (not selecting any)
- 1 orange: 1 way
- 2 oranges: 1 way
- 3 oranges: 1 way
- 4 oranges: 1 way
- 5 oranges: 1 way
- Total ways to select oranges = 5 + 1 = 6 ways.

3. **Calculate the Selection Ways for Mangoes**:
- We can select from 0 to 8 mangoes. This gives us:
- 0 mangoes: 1 way
- 1 mango: 1 way
- 2 mangoes: 1 way
- 3 mangoes: 1 way
- 4 mangoes: 1 way
- 5 mangoes: 1 way
- 6 mangoes: 1 way
- 7 mangoes: 1 way
- 8 mangoes: 1 way
- Total ways to select mangoes = 8 + 1 = 9 ways.

4. **Calculate the Selection Ways for Plantains**:
- We can select from 0 to 7 plantains. This gives us:
- 0 plantains: 1 way
- 1 plantain: 1 way
- 2 plantains: 1 way
- 3 plantains: 1 way
- 4 plantains: 1 way
- 5 plantains: 1 way
- 6 plantains: 1 way
- 7 plantains: 1 way
- Total ways to select plantains = 7 + 1 = 8 ways.

5. **Combine the Selection Ways**:
- The total number of ways to select fruits is the product of the ways to select each type of fruit:
\[ \text{Total Ways} = (\text{Ways to select oranges}) \times (\text{Ways to select mangoes}) \times (\text{Ways to select plantains}) \]
\[ \text{Total Ways} = 6 \times 9 \times 8 \]

6. **Calculate the Total**:
- Now, we compute:
\[ 6 \times 9 = 54 \]
\[ 54 \times 8 = 432 \]

7. **Exclude the Case of Selecting No Fruits**:
- Since we need to exclude the case where no fruit is selected, we subtract 1 from the total:
\[ \text{Final Total Ways} = 432 - 1 = 431 \]

### Final Answer:
The total number of ways to select from the fruits is **431**.