Intro
Combinatorics questions reduce to counting principles: multiply independent choices, add mutually exclusive cases, and use nCr when order does not matter. Write whether order matters before reaching for a formula. Keywords: permutations and combinations, counting principles, HSC combinatorics. NSW Year 12.
Summary
Addition principle for either/or; multiplication for sequential choices. Permutations nPr when order matters; combinations nCr when it does not. Binomial theorem ties to nCr. Extension 2 may add inclusion–exclusion or pigeonhole reasoning.
Combinatorics at Extension 1 depth differs from Extension 2 enrichment — know your syllabus boundary. Counting mistakes in probability often trace back to misapplied nCr versus nPr. Listing small cases validates formulas before exam submission.
Key Points
- Multiplication principle: independent sequential choices multiply.
- Addition principle: disjoint cases add.
- nPr = n!/(n−r)! for ordered selections; nCr = n!/((n−r)!r!) unordered.
- Binomial (x + y)^n expands with coefficients nCr.
- Verify small n by listing — catches formula misuse.
- Structured practice in HSC Combinatorics booklet.
Worked example
Question. How many ways can 5 students sit in a row if two particular students must sit together?
Solution.
- Treat the pair as one block → 4 blocks to arrange: 4! ways.
- The pair within the block: 2! orders.
- Total = 4! × 2! = 24 × 2 = 48.
Answer. 48 arrangements.
Takeaway. 'Together' → block method; remember internal order of the block.
Exam Preparation
Combinatorics errors come from misidentifying order vs unordered. Drill word-problem translation: define a 'choice', decide multiply vs add, then pick nPr or nCr. Link to probability for hybrid questions.
For mixed counting and probability questions, solve the counting part first with nCr or nPr, then divide by total outcomes if uniform probability is stated. Write the sample space size explicitly — examiners award marks for correct setup even if arithmetic slips later.
- Order test. Ask 'would swapping two items create a new outcome?' before nCr.
- Block and gap methods. Practice restriction problems (together, apart, ends).
- Binomial link. Expand (2x − 3)⁵ finding specific term coefficients.
Committee and seating problems are classic HSC wording — translate English into counting actions before calculating. When a question says 'at least one', consider complement counting. Binomial probability and combinatorics overlap; nCr appears inside probability formulas. For Extension 2 enrichment, pigeonhole arguments require clear statement of holes and pigeons — practise the template from the combinatorics booklet. Write whether order matters in one sentence before any calculation.
Mini-FAQ
When do I use nCr vs nPr?
nCr when order does not matter (committees); nPr when order matters (races, passwords with position).
Are circular arrangements in Extension 1?
Extension 2 may include them — check your syllabus booklet; fix one point if rotation is identified.
How does binomial theorem connect?
Coefficient of x^k in (1 + x)^n is nCk — bridges algebra and counting.
Common mistakes to avoid
- Using nCr when order matters (passwords, podiums).
- Forgetting internal arrangements in block method.
- Double-counting overlapping cases.
- Not subtracting forbidden cases from total when easier than direct count.
Combinatorics word-problem template: (1) define a configuration, (2) decide order matters?, (3) multiply or add?, (4) apply nPr or nCr, (5) verify with small n. Write the template at the top of your working in exams — markers see structured thinking. When 'at least one' appears, try complement counting before direct enumeration.
Binomial expansion connects combinatorics to algebra — coefficient questions are high-frequency in Extension 1. Circular arrangements and inclusion–exclusion sit at Extension 2 depth; confirm your syllabus before over-studying enrichment topics.
List all small cases when n = 2, 3 before applying a general counting formula — verification takes thirty seconds and prevents systematic formula errors. Binomial coefficient questions link directly to nCr; state which term you seek before expanding. Extension 2 counting may require inclusion–exclusion; draw a Venn diagram before calculating. Complement counting often simplifies 'at least one' word problems. Permutation vs combination is the first decision in every counting question. Block-and-gap problems appear often in trials — practise both restriction types weekly. The Combinatorics booklet on vumaths.com orders topics from basic counting to Extension 2 enrichment.
Practice on Vu's Maths Hub
Need more practice on this topic? Open the free HSC Combinatorics booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required.
More on Vu's Maths Hub
All booklets are free for personal and school use under the CC BY 4.0 licence.
No comments:
Post a Comment