Number of ways in which 5 prizes be given away to 8 students when elach student is eligible for all the prizes is

Options

A. `5^(8)`
B. `8^(5)`
C. `8xx5!`
D. `5 xx 8!`

Answer

To solve the problem of how many ways 5 prizes can be given away to 8 students when each student is eligible for all the prizes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 5 prizes and 8 students. Each student can win any number of prizes, including none. This means that for each prize, any of the 8 students can be chosen.

2. **Choosing for Each Prize**: For the first prize, we can choose any one of the 8 students. Therefore, there are 8 options for the first prize.

3. **Repeating for Each Prize**: Since the selection of students for each prize is independent of the others, we can repeat the same reasoning for the remaining prizes. For the second prize, we again have 8 options, for the third prize, 8 options, and so on.

4. **Calculating Total Ways**: Since the choices are independent, we multiply the number of choices for each prize. Thus, the total number of ways to distribute the prizes is:
\[ 8 \times 8 \times 8 \times 8 \times 8 = 8^5 \]

5. **Final Calculation**: Now we can calculate \(8^5\):
\[ 8^5 = 32768 \]

### Conclusion:
The total number of ways to give away 5 prizes to 8 students is \(32768\).