Combinatorics — A sum of first-column diagonals

Question

(i) Use Pascal’s identity to 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.Part (i) is Pascal’s identity rearranged.
Simplify the previous line carefully, keeping exact values where possible.Replace each term by a difference of adjacent choose-2 terms.
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: .

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