Variation 1 · answer guide

Combinatorics — Different side counts

Question

Points: 2 on , 3 on , 4 on . How many triangles can be formed from the chosen points?

This practice question was inspired by Question 7 in the 2022 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.Total points ; .
Simplify the previous line carefully, keeping exact values where possible.Collinear exclusions: , , .
Finish the calculation, then check that the result meets the question’s conditions.Triangles: .
Match the result to the choices and state the correct option.Therefore option C is correct.

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

Verification: Subtract only collinear triples on each side.

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