Circular seating with a fixed seat for the host: A host and 8 guests are seated around a round table, but the host must occupy one specific seat. In how many ways can everyone be seated?

Introduction / Context:
In circular permutations, fixing one position removes rotational symmetry. If the host is fixed at a particular labeled seat, the remaining 8 guests can be arranged freely around the table as in a line relative to that fixed reference seat.

Given Data / Assumptions:

• One host assigned to a specific seat.
• 8 other distinct guests to arrange.
• Seats are effectively labeled due to the fixed reference.

Concept / Approach:

• With host fixed, count permutations of the 8 remaining guests around the table.

Step-by-Step Solution:

Verification / Alternative check:
If instead the circle were unlabelled with no fixed person, arrangements would be (9−1)! = 8!. Here, explicitly fixing the host to a particular seat leads to the same count for the remaining guests, confirming 8!.

Why Other Options Are Wrong:

• 9! would overcount by ignoring circular equivalence.
• 6! or 4! are unrelated to 8 free placements.

Common Pitfalls:

• Confusing “host fixed seat” with “host fixed position up to rotation.” Both give 8! for the others.

Final Answer:
8!