There are n numbered seats around a round table. Total number of ways in which `n_(1)(n_(1) lt n)` persons can sit around the round table, is equal to

Options

A. `.^(n)C_(n_(1))`
B. `.^(n)P_(n_(1))`
C. `.^(n)C_(n_(1)-1)`
D. `.^(n)P_(n_(1-1))`

Answer

To find the total number of ways in which \( n_1 \) persons can sit around a round table with \( n \) numbered seats (where \( n_1 < n \)), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have \( n \) seats around a round table and we want to arrange \( n_1 \) persons in these seats. Since the table is round, the arrangement is circular, which affects how we count the arrangements.

2. **Choosing Seats**: First, we need to choose \( n_1 \) seats from the \( n \) available seats. The number of ways to choose \( n_1 \) seats from \( n \) is given by the combination formula:
\[ \binom{n}{n_1} = \frac{n!}{n_1!(n-n_1)!} \]

3. **Arranging Persons**: After choosing the \( n_1 \) seats, we can arrange the \( n_1 \) persons in those seats. The number of ways to arrange \( n_1 \) persons is given by \( n_1! \).

4. **Total Arrangements**: Since the arrangement around a round table is considered the same if rotated, we can use the formula for permutations of \( n_1 \) persons in \( n \) seats. The total number of arrangements is:
\[ \text{Total Ways} = \binom{n}{n_1} \times n_1! = \frac{n!}{(n-n_1)!} \]

5. **Final Expression**: Thus, the total number of ways \( n_1 \) persons can sit around a round table with \( n \) seats is:
\[ \frac{n!}{(n-n_1)!} \]

### Final Answer:
The total number of ways in which \( n_1 \) persons can sit around the round table is:
\[ \frac{n!}{(n-n_1)!} \]