Combinatorics — Gap distribution factor

Question

Three gaps must be filled with people. The number of ways to choose which gap gets person is

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.Any one of the three gaps can be the singleton gap.
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: This is the factor inside .

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