in how many ways can 5 prizes be given away to 7 boys when each boy is eligible for all the prizes ?

Answer

To solve the problem of how many ways 5 prizes can be given away to 7 boys, where each boy is eligible for all the prizes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 5 distinct prizes and 7 boys. Each boy can receive any of the prizes, and there are no restrictions on how many prizes a boy can receive.

2. **Choosing a Prize Recipient**: For each prize, we have 7 options (since any of the 7 boys can receive the prize).

3. **Calculating for Each Prize**:
- For the first prize, there are 7 choices (any of the 7 boys).
- For the second prize, there are again 7 choices.
- This pattern continues for all 5 prizes.

4. **Using the Multiplication Principle**: Since the choice of recipient for each prize is independent of the others, we can multiply the number of choices for each prize:
\[ \text{Total Ways} = 7 \times 7 \times 7 \times 7 \times 7 = 7^5 \]

5. **Calculating \(7^5\)**:
\[ 7^5 = 7 \times 7 \times 7 \times 7 \times 7 = 16807 \]

6. **Conclusion**: Therefore, the total number of ways to distribute the 5 prizes among the 7 boys is \(16807\).

### Final Answer:
The total number of ways to give away 5 prizes to 7 boys is **16807**.

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