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← 2025 HSC Extension 1 paper Question 13(b)

Variation 3 · answer guide

Combinatorics — A Pascal telescoping sum

Question

Evaluate .

This practice question was inspired by Question 13(b) 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.

Step by step

  1. Set up the solution: count a simple unrestricted set first, then add or subtract the arrangements that break the condition.Use .
  2. Simplify the previous line carefully, keeping exact values where possible.The sum telescopes to .
  3. Finish the calculation, then check that the result meets the question’s conditions..
  4. State the final answer clearly in the form the question requested.Final answer:

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.

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