Vu's Maths Hub: HSC Combinatorics: Counting, Permutations, and Binomial Identities

Intro

The HSC Combinatorics booklet is a free Mathematics Extension 1 resource with 60 worked problems covering permutations, combinations, Pascal's triangle, circular arrangements, pigeonhole principle, and proof-style binomial identities. It is written for Extension 1 students wanting extra combinatorics in a classroom-friendly format and designed for structured HSC revision on Vu's Maths Hub.

This deep-dive introduces HSC Combinatorics — browser-readable, aligned with the NESA syllabus.

Summary

The HSC Combinatorics booklet offers a fundamentals review, 60 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Combinatorics — no account required. Use this post to plan how to work through a 82-page booklet efficiently.

What is this booklet?

This booklet is written for Extension 1 students wanting extra combinatorics in a classroom-friendly format.

Focus: permutations, combinations, Pascal's triangle, circular arrangements, pigeonhole principle, and proof-style binomial identities.

Topics covered:

Permutations and combinations
Pascal's triangle
Circular arrangements
Committee selection
Simple probability and expected value
Pigeonhole principle
Geometric counting
Proof-style binomial identities

How to use it:

Read core formulas first; attempt each problem before the solution
Decide selection vs arrangement explicitly
Look for gaps, complementary counting, and balanced groups
Use takeaways for pattern recognition across problems

Approximately 82 pages, CC BY 4.0, readable at HSC Combinatorics.

Key fundamental reviews

Core ideas are embedded in the introduction and early problems. Before Part 1, ensure you can handle the following without notes:

Factorial notation and counting principles
Permutations with and without repetition
Combinations $^{n} C_{r}$ and symmetry
Circular arrangements and fixed-position adjustments
Binomial coefficients and Pascal's triangle
Pigeonhole principle and complementary counting

Why fundamentals matter

Factorial notation and counting principles — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Permutations with and without repetition — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Combinations $^{n} C_{r}$ and symmetry — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Circular arrangements and fixed-position adjustments — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Binomial coefficients and Pascal's triangle — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Students who skip this section often repeat the same algebra errors in Part 2. Treat HSC Combinatorics fundamentals as a closed-book quiz first.

Problems and how to use them

The HSC Combinatorics booklet packs 60 practice problems into roughly 82 pages — well beyond a single textbook chapter. Each item includes worked solutions; many include Takeaways that highlight the method to reuse in exams.

Overall structure

Part 1 — detailed solutions: basic, medium, and advanced counting problems with full solutions.

Part 2 — hint-based fluency: warm-up drills, stretch problems, and challenge corner.

Use Part 1 to learn how complete NSW HSC working is written. Use Part 2 in the fortnight before trials — hints are upside-down so you attempt first.

Part 1 (23 problems)

Advanced (10 problems)

This tier contains 10 problems aimed at extension and synthesis. Representative work includes "Committee selection" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

Committee selection
Pascal's triangle and a telescoping sum
Combining binomial coefficients
Equal numbers of men and women
A binomial identity and double counting capstone
Students and teachers in a circle
Guests who refuse to sit together
Australian Powerball counting
License plates and careful reading
A mixed combinatorics review

Basic (4 problems)

This tier contains 4 problems aimed at foundational fluency. Representative work includes "Four-digit numbers from 1 to 8" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

Four-digit numbers from 1 to 8
Rearranging the letters of COOKED
Choosing three coloured balls
Rearranging the letters of CONDOBOLIN

Medium (9 problems)

This tier contains 9 problems aimed at exam-standard reasoning. Representative work includes "Seating around a circle" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

Seating around a circle
Selections from URGOATED
Forming a Dota 2 e-sport team
Triangles from points on the sides of a triangle
Age-limit check across teams
Expected score with replacement
Minimum votes needed to win
Selecting finalists and awarding places
Using the pigeonhole principle with topic choices

Part 2 (37 problems)

Advanced (17 problems)

This tier contains 17 problems aimed at extension and synthesis. Representative work includes "Comparing coefficients in (1+x)^m+n" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

Comparing coefficients in (1+x)^m+n
Grid paths and binomial identities
A coefficient identity for 4n2n
Repeated draws of green counters
Arrangement formulas for x people
A pair summing to 21
Positive integer solutions
Non-negative integer solutions
Three identities from (1+x)^n
Making an amount of money with Australian denominations
Unlocking a phone by trial and error
Polynomial derivatives and a power-series expansion
Binary strings and a binomial identity
Derangements
A monochromatic triangle on six vertices
A more general form of the binomial theorem
Isomers --- a possible computing project

Basic (9 problems)

This tier contains 9 problems aimed at foundational fluency. Representative work includes "Expanding a binomial surd expression" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

Expanding a binomial surd expression
When is a conjugate surd sum rational?
PINs with numbers and letters
Two coins of the same metal
Probabilities with the four jacks
Counting rectangles in a 5\times 6 grid
How fast does brute force grow?
Repeated derivatives of 1x
Using Pascal's triangle to differentiate x^4

Medium (11 problems)

This tier contains 11 problems aimed at exam-standard reasoning. Representative work includes "Term independent of x" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

Term independent of x
Five-letter words from Mamungkukumpurangkuntjunya
Constant term from a product of two expansions
Adults and children at the cinema
The alternating row constraint
Lines determined by points on a line and a circle
Dota knockout bracket
Dividing a class into equal groups
Triangles from points on a circle
Triangles from a regular octagon and its centre
Pigeons and pigeonholes

Common patterns across the booklet

Mechanics: 1 problem — e.g. "How fast does brute force grow?"
Probability & counting: 11 problems — e.g. "Combining binomial coefficients"
Polynomials: 1 problem — e.g. "Polynomial derivatives and a power-series expansion"
Trigonometry: 10 problems — e.g. "Pascal's triangle and a telescoping sum"
Functions: 2 problems — e.g. "Arrangement formulas for x people"
Other synthesis: 35 problems — e.g. "Committee selection"

Standout and less-seen problem types

These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:

Committee selection: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Pascal's triangle and a telescoping sum: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Combining binomial coefficients: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Equal numbers of men and women: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
A binomial identity and double counting capstone: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Students and teachers in a circle: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.

Working through a large booklet

Read core formulas first; attempt each problem before the solution
Decide selection vs arrangement explicitly
Look for gaps, complementary counting, and balanced groups
Use takeaways for pattern recognition across problems

Timed practice: allow about one minute per mark; write legible structure even when practising alone. Error log: tag mistakes as concept, algebra, or misreading. Rotation: do not camp on advanced tier if basics still slip.

Open HSC Combinatorics and work steadily — 60 problems is a marathon, not a sprint.

Key appendices

Gamma and Beta functions — Enrichment beyond the school syllabus. Use for quick reference or enrichment beyond routine exam questions.

Stars and bars — Distributing identical objects into bins. Use for quick reference or enrichment beyond routine exam questions.

Algorithmic counting — Complexity preview for computer-science minded students. Use for quick reference or enrichment beyond routine exam questions.

Inclusion–exclusion — Counting with overlapping conditions. Use for quick reference or enrichment beyond routine exam questions.

Injective, surjective, bijective — Function-based counting foundations. Use for quick reference or enrichment beyond routine exam questions.

For HSC preparation, prioritise the first one or two appendices; later entries reward curious students but are not required for standard papers.

Key conclusion

Combinatorics rewards careful reading — decide whether a task is arrangement, selection, or probability built from counting before you write a formula.

The booklet's closing section reinforces these habits:

Read core formulas first; attempt each problem before the solution
Decide selection vs arrangement explicitly
Look for gaps, complementary counting, and balanced groups
Use takeaways for pattern recognition across problems

Revisit HSC Combinatorics in the fortnight before trials and redo problems you missed on first pass.

How to study with this booklet

Start with Part 1 basic counting; add medium committee problems; finish with advanced binomial proofs; use Probability booklet for combined questions

General principles:

Closed-book first: attempt without notes, then check fundamentals.
Error log: record concept vs algebra vs reading errors.
Spaced repetition: redo missed questions after 3 and 7 days.
Past papers last: fix weak topics here, then sit full papers timed.

Mini-FAQ

Who is the HSC Combinatorics booklet for?

Extension 1 students wanting extra combinatorics in a classroom-friendly format studying Mathematics Extension 1 under the NSW HSC.

Should I read solutions before attempting problems?

Attempt Part 1 first. Use Part 2 hints only after a genuine try or partial working.

Where can I read the booklet online?

Open HSC Combinatorics on Vu's Maths Hub — free, no account required.

How many problems are in the booklet?

Roughly 60 practice problems across 82 pages, each with worked solutions.

Is this aligned with NESA?

Topics match Mathematics Extension 1 outcomes for permutations, combinations, Pascal's triangle, circular arrangements, pigeonhole principle, and proof-style binomial identities. Confirm scope with your teacher and current NESA documentation.

Common mistakes to avoid

Treating a circular arrangement as a line — divide by $n$ when rotations coincide
Double-counting when conditions overlap — draw cases or use inclusion–exclusion
Using permutations when order does not matter
Forgetting to subtract forbidden cases in "at least one" problems
Rushing to advanced tiers before basic fluency — build foundations first.

Practice on Vu's Maths Hub

Open the free HSC Combinatorics on Vu's Maths Hub — 60 problems with full worked solutions.

Related resources:

How to use Vu's Maths Hub — Combinatorics and probability links
HSC Probability — Counting setups for probability

Permutations and combinations — exam context

In NSW Mathematics Extension 1 examinations, permutations and combinations routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Combinatorics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Combinatorics and locate items that stress permutations and combinations; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Pascal's triangle — exam context

In NSW Mathematics Extension 1 examinations, pascal's triangle routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Combinatorics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Combinatorics and locate items that stress pascal's triangle; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Circular arrangements — exam context

In NSW Mathematics Extension 1 examinations, circular arrangements routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Combinatorics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Combinatorics and locate items that stress circular arrangements; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Committee selection — exam context

In NSW Mathematics Extension 1 examinations, committee selection routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Combinatorics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Combinatorics and locate items that stress committee selection; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Simple probability and expected value — exam context

In NSW Mathematics Extension 1 examinations, simple probability and expected value routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Combinatorics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Combinatorics and locate items that stress simple probability and expected value; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Pigeonhole principle — exam context

In NSW Mathematics Extension 1 examinations, pigeonhole principle routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Combinatorics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Combinatorics and locate items that stress pigeonhole principle; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Geometric counting — exam context

In NSW Mathematics Extension 1 examinations, geometric counting routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Combinatorics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Combinatorics and locate items that stress geometric counting; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Proof-style binomial identities — exam context

In NSW Mathematics Extension 1 examinations, proof-style binomial identities routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Combinatorics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Combinatorics and locate items that stress proof-style binomial identities; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Syllabus alignment

This booklet supports Mathematics Extension 1 under the NESA syllabus. It supplements school instruction with 82 pages of extra exam-style practice — not a replacement for class teaching.

Additional exam advice

When sitting Mathematics Extension 1 exams, allocate time proportional to marks. Practise concise justification in HSC Combinatorics — NSW markers reward clear communication. Reread the booklet conclusion the night before for a habit checklist.

Why Vu's Maths Hub

Vu's Maths Hub hosts every HSC booklet in a continuous, mobile-friendly viewer — zoom, search, no download required. Maintained by Vu Hung Nguyen; CC BY 4.0 for personal and school use.

Wednesday, 1 July 2026

HSC Combinatorics: Counting, Permutations, and Binomial Identities