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 distributing 5 prizes among 8 students where each student is eligible for all the prizes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Prizes and Students**:
We have 5 prizes (let's label them as P1, P2, P3, P4, and P5) and 8 students (S1, S2, S3, S4, S5, S6, S7, and S8).

2. **Determine Choices for Each Prize**:
For each prize, any of the 8 students can receive it. This means that for Prize P1, there are 8 choices (any one of the 8 students can receive it).

3. **Apply the Multiplication Principle**:
Since the choice of student for each prize is independent of the others, we can use the multiplication principle. This means we multiply the number of choices for each prize.

4. **Calculate the Total Number of Ways**:
- For Prize P1, there are 8 choices.
- For Prize P2, there are also 8 choices.
- For Prize P3, there are again 8 choices.
- For Prize P4, there are 8 choices.
- For Prize P5, there are 8 choices.

Therefore, 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 compute \(8^5\):
\[ 8^5 = 32768 \]

Thus, the total number of ways to distribute the 5 prizes among the 8 students is **32768**.