In how many ways can 5 prizes be given away to 7 students when each student is eligible for every prize ?

Options

A. `7^(5)`
B. `5^(7)`
C. 7!
D. 5!

Answer

To solve the problem of distributing 5 prizes among 7 students where each student is eligible for every prize, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 5 distinct prizes and 7 students. Each student can receive any of the prizes.

2. **Choosing a Prize**: For the first prize, we can choose any one of the 7 students to receive it. So, there are 7 possible choices for the first prize.

3. **Continuing for All Prizes**: The same logic applies to the second prize. We again have 7 choices, as any of the 7 students can receive this prize as well.

4. **Repeating for All Prizes**: This process continues for all 5 prizes. Each prize can be given to any of the 7 students, so for each of the 5 prizes, we have 7 choices.

5. **Calculating Total Ways**: Since the choices for each prize are independent, we multiply the number of choices together. Therefore, the total number of ways to distribute the prizes is:
\[ 7 \times 7 \times 7 \times 7 \times 7 = 7^5 \]

6. **Final Calculation**: Now we can calculate \( 7^5 \):
\[ 7^5 = 16807 \]

Thus, the total number of ways to distribute the 5 prizes among 7 students is **16807**.