There are fifty persons among whom 2 are brothers. The number of ways they can be arranged in a circle, if there is exactly one person between the two brothers, is

Options

A. `2xx48!`
B. 12
C. 360
D. `7xx8!`

Answer

To solve the problem of arranging 50 persons in a circle with exactly one person between two brothers, we can follow these steps:

### Step-by-Step Solution:

1. **Fix the Brothers' Positions**:
- Let's denote the two brothers as B1 and B2. To simplify the arrangement in a circle, we can fix one brother (say B1) at a specific position. This is a common technique in circular permutations to eliminate the effect of rotations.

2. **Choose the Person to Sit Between the Brothers**:
- Since we want exactly one person between B1 and B2, we need to select one person from the remaining 48 persons (50 total - 2 brothers = 48). The number of ways to choose one person from 48 is given by:
\[ \text{Ways to choose 1 person} = \binom{48}{1} = 48 \]

3. **Arrange the Brothers**:
- Once we have chosen a person (let's call them P) to sit between B1 and B2, we can arrange B1 and B2 in two ways: either B1 sits to the left of P or B2 sits to the left of P. Thus, there are:
\[ \text{Ways to arrange B1 and B2} = 2! = 2 \]

4. **Arrange the Remaining Persons**:
- After placing B1, B2, and the chosen person P, we have 47 persons left (50 total - 3 already placed). These 47 persons can be arranged in a circle. The number of ways to arrange n persons in a circle is given by \((n-1)!\). Therefore, the number of ways to arrange the remaining 47 persons is:
\[ \text{Ways to arrange 47 persons} = 47! \]

5. **Combine All the Arrangements**:
- Now, we combine all the arrangements calculated in the previous steps. The total number of ways to arrange the 50 persons in a circle with exactly one person between the two brothers is:
\[ \text{Total arrangements} = \text{Ways to choose P} \times \text{Ways to arrange B1 and B2} \times \text{Ways to arrange remaining 47 persons} \]
\[ = 48 \times 2 \times 47! \]

6. **Final Expression**:
- Thus, the final expression for the number of ways to arrange the 50 persons is:
\[ = 2 \times 48 \times 47! \]

### Final Answer:
The number of ways they can be arranged in a circle, if there is exactly one person between the two brothers, is:
\[ \boxed{2 \times 48 \times 47!} \]