Combinatorics — Restricted circular seating

Question

Seven people sit around a round table. Two specified people must not sit together. How many arrangements?

This practice question was inspired by Question 13(b) in the 2025 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: count a simple unrestricted set first, then add or subtract the arrangements that break the condition.Unrestricted circular arrangements: .
Simplify the previous line carefully, keeping exact values where possible.Together: treat the pair as a block, giving circular arrangements and internal orders.
Finish the calculation, then check that the result meets the question’s conditions..
State the final answer clearly in the form the question requested.Final answer:

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: The restricted count is smaller than and non-negative.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.