Combinatorics — Analogous three-term merge

Question

Find one pair such that .

This practice question was inspired by Question 12(d) in the 2023 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.Pascal: .
Simplify the previous line carefully, keeping exact values where possible.Symmetry: .
Finish the calculation, then check that the result meets the question’s conditions.Hence LHS by Pascal again.
State the final answer clearly in the form the question requested.One solution: , (or ).

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

Verification: Pascal, then symmetry, then Pascal — same three moves as the 2023 solution.

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