Combinatorics | HSC Extension 1

Assessment evidence

What these examples cover

These examples show assessable skills, not an official HSC topic weighting or prediction. The 2024 NSW syllabuses have their first HSC examination in 2027; students sitting the 2026 HSC continue under the applicable 2017 syllabus.

Teacher check

Teacher verification note: check the mathematics, notation, permitted technology and syllabus fit for your cohort before classroom or assessment use.

3 deterministic examples

Combinatorics questions with worked solutions

Each question is generated from an existing curriq QDSL archetype at a fixed seed. Marks and catalog content mappings travel with the generated question.

Worked example 1

2 marksext1-a1-2

A fifth-power binomial Expand and simplify

(4-x)^5

Part a · 2 marks

Give the fully simplified polynomial.

Worked solution

Answer: (((1024 - (1280 x)) + (640 x^2)) - (160 x^3)) + (20 x^4) - x^5

Binomial Coefficients: Use coefficients $1,5,10,10,5,1$ with alternating signs.
Collect Polynomial: (((1024 - (1280 x)) + (640 x^2)) - (160 x^3)) + (20 x^4) - x^5

Catalog reference: ext1_jr25_binomial_fifth_power_expansion · seed 41001

Worked example 2

2 marksext1-a1-2

A coefficient after substitution Find the coefficient of

y^3

(2-2y+y^2)^5

Part a · 2 marks

Find the coefficient.

Worked solution

Answer: -640

Coefficient From Square: $-202^32$ .
Coefficient From Cube: $-102^22^3$ ; add the two contributions.

Catalog reference: ext1_jr25_composite_binomial_y3_coefficient · seed 41001

Worked example 3

1 markext1-a1-1

Repeated-letter arrangements A word contains 7 identical A's, two identical B's, two identical R's, one C and one D. How many distinct linear arrangements are possible?

Part a · 1 mark

Find the number of arrangements of all 13 letters.

Worked solution

Answer: 308880

Multiset Permutation: $(13!)/(7!2!2!)$ .

Catalog reference: ext1_pem_2025_q11c_i_repeated_letter_arrangements · seed 41001

Content mapping and common errors

Combinatorics content identifiers

ME-A1

Working with Combinatorics

Apply the multiplication principle, permutations, and combinations; expand using the binomial theorem.

Common student mistakes

·Using permutations where combinations were required (or vice versa)
·Forgetting to account for identical objects in permutation problems
·Incorrect general term index in the binomial expansion

Source and methodology

Examples are deterministic curriq catalog generations from existing QDSL archetypes at the published fixed seed. The generator copies only student-facing prompts, mark allocations, topic mappings, display tables or options, checked answers and marking-path results into this resource.

Review the official NSW Curriculum source ↗

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What these examples cover

Combinatorics questions with worked solutions

Worked example 1

Worked example 2

Worked example 3

Combinatorics content identifiers

Working with Combinatorics

Source and methodology

Other Extension 1 topics

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