The number of ways in which 5 different prizes can be distributed amongst 4 persons if each is entitled to receive at most 4 prizes is:

Options

A. 1020
B. 620
C. `5^(4)-5`
D. `4^(5)-4`

Answer

To solve the problem of distributing 5 different prizes among 4 persons, where each person can receive at most 4 prizes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 5 different prizes (let's label them as P1, P2, P3, P4, and P5) and 4 persons (let's label them as A, B, C, and D). Each prize can go to any of the 4 persons.

2. **Total Distribution Without Restrictions**: If there were no restrictions on the number of prizes each person could receive, each of the 5 prizes could be given to any of the 4 persons. Therefore, the total number of ways to distribute the prizes would be calculated as:
\[ 4^5 \]
This is because for each prize, there are 4 choices (one for each person), and since there are 5 prizes, we multiply the choices.

3. **Calculating \(4^5\)**:
\[ 4^5 = 1024 \]
Thus, there are 1024 ways to distribute the prizes without any restrictions.

4. **Applying the Restriction**: However, we have the restriction that each person can receive at most 4 prizes. This means we need to subtract the cases where at least one person receives all 5 prizes.

5. **Counting Invalid Cases**: If one person (say A) receives all 5 prizes, the other three persons (B, C, D) receive none. Since there are 4 persons, there are 4 such invalid cases (one for each person receiving all prizes):
- A gets all 5 prizes
- B gets all 5 prizes
- C gets all 5 prizes
- D gets all 5 prizes

6. **Subtracting Invalid Cases**: Therefore, we need to subtract these 4 invalid cases from the total number of unrestricted distributions:
\[ 1024 - 4 = 1020 \]

7. **Final Answer**: The number of ways to distribute the 5 different prizes among 4 persons, with each person receiving at most 4 prizes, is:
\[ \boxed{1020} \]