There are n seats round a table numbered `1,2, 3,..., n`. The number of ways in which `m(<=n)` persons can take seats is

Options

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

Answer

To solve the problem of finding the number of ways in which `m` persons can take seats at a round table with `n` seats, we can follow these steps:

### Step 1: Choose `m` Seats from `n` Seats
Since the seats are arranged in a circle, we first need to choose `m` seats from the `n` available seats. The number of ways to choose `m` seats from `n` seats is given by the combination formula:

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

### Step 2: Arrange `m` Persons in the Chosen Seats
After choosing the `m` seats, we need to arrange the `m` persons in these seats. The number of ways to arrange `m` persons is given by the factorial of `m`:

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

### Step 3: Calculate the Total Number of Ways
To find the total number of ways in which `m` persons can take seats, we multiply the number of ways to choose the seats by the number of ways to arrange the persons:

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

### Final Answer
Thus, the total number of ways in which `m` persons can take seats at a round table with `n` seats is:

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

---