Sixteen persons including a host are to take their seats in a row for a group photograph. If two particular persons must always sit on either side of the host, in how many different seating arrangements is this possible?

Introduction / Context:
This is a seating arrangement problem involving a linear row and a special condition about relative positions. We have a host and two particular guests who must always sit immediately on either side of the host. We must count how many different linear arrangements satisfy this constraint when all 16 persons are lined up for a photograph.

Given Data / Assumptions:

There are 16 persons in total, including the host and two particular guests.

Everyone is seated in a single row of 16 seats.

The two particular persons must sit immediately to the left and right of the host, in any order.

The remaining 13 persons can be seated anywhere in the other seats.

Concept / Approach:
We treat the host and the two special persons as a block of three adjacent seats, where the host is in the middle and the two special persons occupy the ends. This block can appear in several positions along the row, and the order of the two special persons around the host can also vary. Once we place this block, we arrange the remaining people in the remaining seats.

Step-by-Step Solution:
Step 1: Consider the three-person block consisting of the host in the middle and the two special guests on either side.Step 2: Within this block, the two special guests can be arranged on the left and right of the host in 2 ways.Step 3: Along the row of 16 seats, this three-seat block can start at positions 1 through 14, so there are 14 possible locations for the block.Step 4: The remaining 13 seats are to be filled by the remaining 13 people. They can be arranged in 13! ways.Step 5: Total arrangements = 14 (positions for the block) * 2 (orders of the two special guests within the block) * 13! (arrangements of the others) = 14! * 2.

Verification / Alternative check:
Another way is to first place the host (16 choices), then place the two special guests on adjacent sides if possible. However, edge positions complicate counting and lead back to a similar block-based reasoning. The block approach is cleaner and confirms the answer 14! × 2.

Why Other Options Are Wrong:
16! × 2 counts all permutations of 16 people and doubles them, ignoring the adjacency constraint around the host.

18! × 2 and 14! do not reflect the correct combination of choices for block placement and internal arrangement of people.

Common Pitfalls:
Students sometimes forget to consider that the three-person block can slide along the row, or they forget the factor of 2 for the interchange of the two special guests. Others mistakenly treat the situation as circular seating or ignore that the host must be in the middle of the block. Carefully identifying the block and then counting its placements and internal permutations avoids these issues.

Final Answer:
The number of valid seating arrangements is 14! × 2.