HSC Combinatorics: Study Guide & Outline

The introduction sets the stage for the booklet, outlining its project overview, target audience (HSC Extension 1 and 2 students), and guidance on how to use the resource effectively. It recaps essential Extension 1 knowledge: factorials, permutations ($P(n,r)$), combinations ($\binom{n}{r}$), and the fundamental counting principle. Core formulas and ideas are reviewed, including the binomial theorem $(x+y)^n=\sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$, binomial probability $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$, Pascal's triangle identities (e.g., $\binom{n}{k}+\binom{n}{k-1}=\binom{n+1}{k}$), and the pigeonhole principle (especially with remainders modulo $n$). The section also instills problem-solving habits: reading carefully, listing systematically, checking for overcounting, and verifying boundary conditions—skills that are reinforced throughout the booklet.

This section presents fully worked, detailed solutions to a curated set of 23 problems, organized into three tiers: Basic (Problems 2.1–2.4), Medium (Problems 2.5–2.13), and Advanced (Problems 2.14–2.23). Basic problems solidify fundamentals: forming four-digit numbers from digits 1–8, rearranging letters of 'COOKED' and 'CONDOBOLIN' with indistinguishable letters, and selecting coloured balls. Medium problems tackle circular seating arrangements, selections from 'URGOATED', forming a Dota 2 team with role assignments, counting triangles from points on a triangle's sides, age-limit constraints, expected scores with replacement, minimum votes to win, selecting finalists and awarding places, and applying the pigeonhole principle to topic choices. Advanced problems explore committee selection with restrictions, Pascal's triangle and telescoping sums, binomial coefficient identities (e.g., using double counting), equal gender distributions, students and teachers around a circle, guests refusing to sit together, Powerball counting, license plate enumeration, and a mixed review. Each solution models clear casework, complementary counting, inclusion-exclusion where needed, and identification of common pitfalls like double counting or order misinterpretation, thereby building rigorous combinatorial thinking.

This extensive bank offers over 30 problems across three levels: Warm-Up Drills (3.1–3.9), Stretch Problems (3.10–3.20), and Challenge Corner (3.22–3.38). Warm-up drills reinforce binomial expansions with surds ($(\sqrt{a}+\sqrt{b})^n$), rationalizing surd sums, PIN generation with alphanumeric characters, coin selections with constraints, probability with playing cards (four jacks), counting rectangles in an $m\times n$ grid, exponential growth in brute-force counting, and using Pascal's triangle to differentiate $x^4$. Stretch problems involve finding the term independent of $x$, constant terms from products of expansions, seating adults and children with alternating rows, lines determined by points on a line and a circle, Dota knockout brackets, dividing a class into equal groups, triangles from points on a circle and a regular octagon with its centre, and pigeonhole principle refinements. Challenge Corner problems are significantly more advanced: grid paths and binomial identities, repeated draws without replacement, arrangement formulas for $x$ people, integer solutions (positive and non-negative) via stars and bars, combinatorial identities from $(1+x)^n$, making change with Australian denominations, unlocking a phone by trial and error, polynomial derivatives and power series, binary strings and identities, derangements, monochromatic triangles (Ramsey theory), the generalized binomial theorem $(1+x)^\alpha$, isomer counting (chemistry application), and fractional parts of binomial expansions. This tier pushes students toward Extension 2 and beyond, fostering proof skills and creative applications of combinatorial principles.

The conclusion synthesizes the key strategies and mathematical insights developed throughout the booklet. It emphasizes the importance of systematic problem analysis, the power of combinatorial identities, and the value of checking and verifying answers. Students are encouraged to reflect on the habits formed—such as careful reading, case enumeration, and complementary counting—and to continue practicing these skills. The conclusion may also point toward further explorations in combinatorics and discrete mathematics, reinforcing that the booklet serves as a solid foundation for Extension 2 and beyond.

The appendices provide optional enrichment material that extends beyond the HSC syllabus. Topics include the Gamma and Beta functions (continuous generalizations of factorials, $\Gamma(n)=(n-1)!$), the stars and bars method for distributing indistinguishable objects into distinct bins ($\binom{n+k-1}{k-1}$), algorithmic counting and complexity (e.g., Big-O notation and decidability), the inclusion-exclusion principle for counting elements with overlapping properties ($|A\cup B\cup C| = |A|+|B|+|C| -|A\cap B| - \dots + |A\cap B\cap C|$), precise definitions of injective, surjective, and bijective functions, and the multinomial theorem $(x_1+\cdots+x_m)^n = \sum \frac{n!}{k_1!\cdots k_m!} x_1^{k_1}\cdots x_m^{k_m}$. These sections equip students with vocabulary and tools for deeper study in combinatorics, probability, and theoretical computer science.