Combinatorics — Linear vs circular teachers

Question

If the distinct teachers were instead arranged in a line (not a circle), their arrangements would number

This practice question was inspired by Question 10 in the 2023 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.Linear arrangements of distinct people: .
Finish the calculation, then check that the result meets the question’s conditions.Circular arrangements would be .
Match the result to the choices and state the correct option.Therefore option A is correct.

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

Verification: The exam multiplies by , combining gap choice with circular teacher seats.

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