20 persons were invited for a party. The number of ways in which they and the host can be seated at a circular table such that two particular persons can be seated on either side of the host is

Options

A. 20!
B. 19!
C. 2(18!)
D. 18!

Answer

To solve the problem of seating 20 persons and a host at a circular table such that two particular persons can be seated on either side of the host, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Number of People**:
We have 20 invited persons and 1 host, making a total of 21 persons.

2. **Fix the Host's Position**:
In circular arrangements, we can fix one person's position to eliminate the effect of rotations. Here, we fix the host's position.

3. **Identify the Positions for the Two Particular Persons**:
Once the host's position is fixed, there are two seats directly next to the host. We want the two particular persons (let's call them A and B) to occupy these two seats.

4. **Arrange Persons A and B**:
Since A and B can be seated on either side of the host, there are 2 ways to arrange A and B in the two seats next to the host:
- A on the left and B on the right
- B on the left and A on the right

5. **Count the Remaining Persons**:
After seating A and B, there are 18 remaining persons (20 total - 2 for A and B).

6. **Arrange the Remaining Persons**:
The remaining 18 persons can be seated in the remaining 18 seats. The number of ways to arrange these 18 persons is given by \(18!\) (18 factorial).

7. **Calculate the Total Arrangements**:
The total number of arrangements is the product of the arrangements of A and B and the arrangements of the remaining persons:
\[ \text{Total Arrangements} = 2 \times 18! \]

### Final Answer:
The total number of ways in which the 20 persons and the host can be seated at a circular table such that the two particular persons can be seated on either side of the host is \(2 \times 18!\).

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