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

Options

A. 18!
B. 2!18!
C. 19!
D. None of these

Answer

To solve the problem of arranging 20 people around a circle such that exactly one person is between two sisters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total People**: We have 20 people in total, out of which 2 are sisters.

2. **Fix the Circle**: When arranging people in a circle, we can fix one person to eliminate the circular permutations. Thus, we can think of arranging the remaining people around this fixed point.

3. **Consider the Arrangement of Other People**: If we take out the two sisters, we are left with 18 other people. We can arrange these 18 people in a circle. The number of ways to arrange \( n \) people in a circle is given by \( (n-1)! \). Therefore, the number of ways to arrange 18 people in a circle is:
\[ 17! \text{ (since } 18 - 1 = 17\text{)} \]

4. **Positioning the Sisters**: We need to place the two sisters such that there is exactly one person between them. When we arrange the 18 people, there will be 18 gaps created between them (including the gap before the first person and after the last person).

5. **Choosing the Gaps**: We can choose any one of these 18 gaps to place the first sister. The second sister must then go into the gap that is two positions away from the first sister (to ensure there is one person between them).

6. **Arranging the Sisters**: The two sisters can be arranged in the chosen gaps in 2 different ways (Sister A in gap 1 and Sister B in gap 2, or Sister B in gap 1 and Sister A in gap 2). This gives us:
\[ 2! = 2 \text{ ways} \]

7. **Final Calculation**: Now, we combine the arrangements of the 18 people and the arrangements of the sisters:
\[ \text{Total arrangements} = 17! \times 18 \times 2 \]
Here, \( 18 \) is the number of ways to choose the gap for the first sister.

8. **Simplifying the Expression**: Thus, the total number of arrangements can be expressed as:
\[ 2 \times 18 \times 17! = 36 \times 17! \]

### Conclusion:
The total number of ways to arrange the 20 people around a circle such that there is exactly one person between the two sisters is:
\[ 36 \times 17! \]