Number of ways in which n different prizes can be distributed among m-persons `(m lt n) ` if each is entitled to receive atmost (n-1) prizes is

Options

A. `n^(m) - n`
B. `m^(n)`
C. mn
D. `m^(n) - m`

Answer

To solve the problem of distributing \( n \) different prizes among \( m \) persons (where \( m < n \)) such that each person can receive at most \( n-1 \) prizes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have \( n \) different prizes and \( m \) persons. Each person can receive a maximum of \( n-1 \) prizes. This means that no person can receive all \( n \) prizes.

2. **Total Ways Without Restrictions**:
If there were no restrictions, each prize can be given to any of the \( m \) persons. Therefore, the total number of ways to distribute \( n \) prizes would be \( m^n \).

3. **Applying the Restriction**:
Since each person can receive at most \( n-1 \) prizes, we need to exclude the cases where at least one person receives all \( n \) prizes.

4. **Counting Invalid Cases**:
If one specific person receives all \( n \) prizes, there are \( m \) ways to choose that person. The remaining \( m-1 \) persons can receive none of the prizes. Thus, the number of invalid distributions (where one person gets all prizes) is \( m \).

5. **Calculating Valid Distributions**:
Therefore, the valid distributions can be calculated as:
\[ \text{Valid Distributions} = m^n - m \]

6. **Generalizing the Formula**:
Since we have established that \( m < n \) and each person can receive at most \( n-1 \) prizes, the final formula for the number of ways to distribute \( n \) different prizes among \( m \) persons, where each can receive at most \( n-1 \) prizes, is:
\[ \text{Number of ways} = m^n - m \]

### Final Answer:
The number of ways in which \( n \) different prizes can be distributed among \( m \) persons (where \( m < n \)) if each is entitled to receive at most \( n-1 \) prizes is:
\[ m^n - m \]