Combinatorics — A choose-three diagonal

Question

(i) Show .

(ii) Hence evaluate .

This practice question was inspired by Question 13(e) 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.Rearrange Pascal’s identity with the lower index .
Simplify the previous line carefully, keeping exact values where possible.Replace every choose-3 term by an adjacent difference.
Simplify the previous line carefully, keeping exact values where possible.The sum telescopes to .
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: (ii) .

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

Verification: Direct values confirm the result.

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