Question

20 persons are invited for a party. the number of ways in which they and the host can be seated at a circular table y two partocular person can be seated on either side og the host is qual to 

• A. 2.(18)!
• B. 18!.3!
• C. 19!.2!
• D. none of the above

Answer 1

The Correct Option is A

To solve the problem, we need to determine the number of seating arrangements when 20 guests and 1 host are seated around a circular table, with the condition that two particular persons must sit on either side of the host.

1. Understanding the Problem:
- Total persons = 20 guests + 1 host = 21 persons
- Arrangement is circular with 2 specific guests (let's call them G₁ and G₂) required to sit adjacent to the host (H)

2. Fixing the Host's Position (Circular Permutation):
For circular arrangements, we fix one position to eliminate equivalent rotations:
- Fix the host in a specific seat: H = fixed position
- This converts the problem to linear arrangement relative to H

3. Arranging the Two Special Guests:
The condition requires:
- G₁ must sit immediately to H's left or right
- G₂ must sit on the opposite side
Possible arrangements:
1. (G₁-H-G₂)
2. (G₂-H-G₁)
Number of arrangements for these two guests: $2! = 2$

4. Arranging Remaining Guests:
- After placing H, G₁, and G₂, we have 18 seats left
- 18 guests can be arranged in these seats in: $18!$ ways

5. Calculating Total Arrangements:
Total valid arrangements = (Arrangements of G₁/G₂) × (Arrangements of others)
$= 2 × 18!$

Verification:
- Host's position is fixed (1 way)
- 2 choices for adjacent guests' positions
- 18! arrangements for others
- Total matches our calculation

Final Answer:
The number of valid seating arrangements is $\boxed{2 × 18!}$.