There are 20 persons including two brothers. In how many ways can they be arranged on a round table if: There is exactly one person between the two brothers.

Options

A. a. `(14!)^2`
B. b. `(14!)`
C. c. `2.(18!)`
D. d. none of these

Answer

To solve the problem of arranging 20 persons, including two brothers, around a round table with the condition that exactly one person sits between the two brothers, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arrangement of Persons**:
- We have a total of 20 persons, which includes 2 brothers.
- Since we are arranging them around a round table, we need to consider the arrangement in a circular manner.

2. **Fixing the Brothers**:
- To simplify the arrangement, we can treat the two brothers as a single unit or block since we want exactly one person to sit between them.
- Therefore, we can visualize the arrangement as: Brother 1, Person X (the one sitting between them), Brother 2.

3. **Selecting the Person Between the Brothers**:
- There are 18 other persons (20 total - 2 brothers = 18 persons) who can occupy the position between the two brothers.
- We can choose any one of these 18 persons to sit between the two brothers.

4. **Arranging the Remaining Persons**:
- After placing the two brothers and the selected person in between, we have 17 remaining persons (20 total - 3 persons already placed = 17 persons).
- These 17 persons can be arranged around the round table. The number of ways to arrange n distinct objects in a circle is (n-1)!.
- Therefore, the number of ways to arrange the remaining 17 persons is \( 17! \).

5. **Arranging the Brothers**:
- The two brothers can be arranged in 2 different ways (Brother 1 can be on the left or Brother 2 can be on the left).

6. **Calculating the Total Arrangements**:
- The total arrangements can be calculated as follows:
\[ \text{Total arrangements} = (\text{Ways to choose the person between brothers}) \times (\text{Ways to arrange remaining persons}) \times (\text{Ways to arrange brothers}) \]
\[ = 18 \times 17! \times 2 \]

7. **Final Answer**:
- Thus, the total number of ways to arrange the 20 persons with exactly one person between the two brothers is:
\[ 36 \times 17! \]