There are n numbered seats around a round table. In how many ways can m (< n) persons sit around the round table.

Options

A. `""^(n)C_(m)*m!`
B. `""^(n)C_(m)*(m-1)t`
C. `(""^(n)C_(m)*(m-1)!)/(2)`
D. none

Answer

To solve the problem of how many ways m persons can sit around a round table with n numbered seats (where m < n), we can follow these steps:

### Step-by-Step Solution:

1. **Choosing Seats**:
Since we have n seats and we need to choose m seats for the m persons, we can use the combination formula to select the seats. The number of ways to choose m seats from n is given by:
\[ \binom{n}{m} \]

2. **Arranging Persons**:
After selecting m seats, we need to arrange m persons in those chosen seats. The number of ways to arrange m persons is given by:
\[ m! \]

3. **Total Arrangements**:
To find the total number of ways m persons can sit around the round table, we multiply the number of ways to choose the seats by the number of ways to arrange the persons:
\[ \text{Total Ways} = \binom{n}{m} \times m! \]

### Final Answer:
Thus, the total number of ways m persons can sit around a round table with n seats is:
\[ \binom{n}{m} \times m! \]