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 around a circle such that there is exactly one person between two brothers, we can follow these steps:

### Step-by-Step Solution:

1. **Fix the Circle**: In circular permutations, we can fix one person to eliminate the effect of rotations. We can fix one of the brothers (let's say Brother 1, B1) in one position.

2. **Positioning the Second Brother**: Since we want exactly one person between the two brothers, Brother 2 (B2) must be placed two positions away from B1. This means there is one person between them.

3. **Choosing the Person Between the Brothers**: We have 18 remaining persons (since we have already fixed B1 and B2). We need to choose one person from these 18 to sit between B1 and B2. The number of ways to choose this person is \( C(18, 1) = 18 \).

4. **Arranging the Brothers and the Chosen Person**: Once we have chosen the person (let's call them M), we can arrange B1, M, and B2 in two ways: either B1, M, B2 or B2, M, B1. Therefore, there are \( 2! = 2 \) ways to arrange these three.

5. **Arranging the Remaining Persons**: After placing B1, M, and B2, we have 17 remaining persons left to arrange. The number of ways to arrange these 17 persons is \( 17! \).

6. **Calculating the Total Arrangements**: Now, we can combine all these calculations to find the total number of arrangements:
\[ \text{Total arrangements} = C(18, 1) \times 2! \times 17! = 18 \times 2 \times 17! \]

7. **Simplifying the Expression**: We can simplify this further:
\[ 18 \times 2 \times 17! = 36 \times 17! \]

### Final Answer:
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 \( 36 \times 17! \).