There are 20 persons among whom are two brothers. The number of ways in which we can arrange them around a circle so that there is exactly one person between the two brothers, is

Options

A. 18!
B. `17!xx2!`
C. `18!xx2!`
D. 20!

Explanation

To solve the problem of arranging 20 persons in a circle such that there is exactly one person between two brothers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the two brothers**: Let's denote the two brothers as B1 and B2.

2. **Fix the arrangement of brothers**: Since we want exactly one person between the two brothers, we can visualize the arrangement as follows:
- Place B1, then a person M (who is not a brother), and then B2.
- Alternatively, we can also have B2, then M, and then B1. This gives us two configurations:
- Configuration 1: B1 - M - B2
- Configuration 2: B2 - M - B1

3. **Choose the person M**: Since B1 and B2 are already chosen, we have 18 remaining persons to choose from for M. Therefore, the number of ways to choose M is:
\[ \text{Number of ways to choose M} = 18 \]

4. **Fix the arrangement in a circle**: When arranging people in a circle, we can fix one person to eliminate identical arrangements due to rotation. In this case, we can fix M. After fixing M, we have:
- 18 persons total (including M) minus 1 (since M is fixed), which leaves us with 17 persons to arrange.

5. **Calculate the arrangements**: The number of ways to arrange the remaining 17 persons in a circle is given by:
\[ \text{Number of arrangements} = (17 - 1)! = 16! \]

6. **Combine the configurations and arrangements**: Since we have two configurations (B1-M-B2 and B2-M-B1) and we can choose M in 18 ways, the total number of arrangements is:
\[ \text{Total arrangements} = 2 \times 18 \times 16! \]

### Final Calculation:
Thus, the total number of ways to arrange the 20 persons around a circle such that there is exactly one person between the two brothers is:
\[ \text{Total ways} = 2 \times 18 \times 16! \]