There are number of seats and m number of people have to be seated, then how many ways are possible to do this `(mltn)`?

Options

A. `""^(n)P_(m)`
B. `""^(n)C_(m)`
C. `""^(n)C_(n)xx(m-1)!`
D. `""^(n-1)P_(m-1)`

Answer

To solve the problem of seating \( m \) people in \( n \) seats, where \( m < n \), we can follow these steps:

### Step 1: Choose the Seats
First, we need to select \( m \) seats from the \( n \) available seats. The number of ways to choose \( m \) seats from \( n \) seats can be calculated using the combination formula:

\[ \text{Number of ways to choose seats} = \binom{n}{m} \]

### Step 2: Arrange the People
After selecting \( m \) seats, we can arrange the \( m \) people in those chosen seats. The number of ways to arrange \( m \) people is given by the factorial of \( m \):

\[ \text{Number of ways to arrange people} = m! \]

### Step 3: Combine the Two Steps
To find the total number of ways to seat \( m \) people in \( n \) seats, we multiply the number of ways to choose the seats by the number of ways to arrange the people:

\[ \text{Total ways} = \binom{n}{m} \times m! \]

### Step 4: Simplify the Expression
The expression \(\binom{n}{m} \times m!\) can be simplified to the permutation formula \( P(n, m) \):

\[ P(n, m) = \frac{n!}{(n - m)!} \]

Thus, the total number of ways to seat \( m \) people in \( n \) seats is:

\[ P(n, m) = \frac{n!}{(n - m)!} \]

### Final Answer
The total number of ways to seat \( m \) people in \( n \) seats is:

\[ P(n, m) = nPm \]

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