Combinatorics — A committee with a condition

Question

Choose a committee of 4 from 9 people if two specified people cannot both serve.

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.All committees: .
Simplify the previous line carefully, keeping exact values where possible.Committees containing both specified people: choose the other two from seven, .
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: Directly count committees containing zero or one of the specified people to confirm.

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