Combinatorics — A short hockey-stick sum

Question

(i) Derive from Pascal’s identity.

(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 to obtain the difference in part (i).
Simplify the previous line carefully, keeping exact values where possible.Write each term as .
Simplify the previous line carefully, keeping exact values where possible.The middle terms cancel.
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 calculation gives .

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