Units 1 & 2 (Permutations and Combinations)
Focus on foundational counting principles, including the difference between arrangements (permutations) and selections (combinations), with and without repetition. Emphasis is placed on using factorial notation, solving real-world problems involving ordered and unordered selections, and identifying when to apply each counting technique.

Unit 1: Fundamental Principles of Counting

1.1 Basic Counting Principles

• Multiplication and Addition Rules: Understanding when to multiply vs. add outcomes in compound events.

• Tree Diagrams and Tables: Visual representation for small cases.

1.2 Factorials and Permutations

• Definition of n!: Interpreting and computing factorials.

• Permutations Without Repetition: P(n,r)=n!(n−r)!P(n, r) = \frac{n!}{(n - r)!}P(n,r)=(n−r)!n!​, applied to ordered arrangements.

• Permutations With Repetition: Identifying identical items (e.g., in words like “LEVEL”) and adjusting the total count.

1.3 Combinations and Binomial Coefficients

• Combinations Without Repetition: C(n,r)=n!r!(n−r)!C(n, r) = \frac{n!}{r!(n - r)!}C(n,r)=r!(n−r)!n!​, used for unordered selections.

• Combinations With Repetition: Understanding the "stars and bars" method for distributing identical items.

• Applications: Team selection, groupings, and distributing items into sets.

Unit 2: Applied Combinatorics and Problem Solving

2.1 Comparing Permutations and Combinations

• Decision-Making: Recognizing whether order matters and selecting the appropriate method.

• Mixed Problems: Scenarios requiring both permutation and combination reasoning.

2.2 Advanced Arrangements and Restrictions

• Conditional Arrangements: e.g., placing certain people together or apart.

• Circular Permutations: Accounting for rotational symmetry: (n−1)!(n - 1)!(n−1)!.

• Dividing into Groups: Using combinations to partition sets into equal or unequal groups.

2.3 Probability Connections

• Uniform Probability Contexts: Using combinations and permutations to calculate probabilities.

• Non-Uniform Scenarios: Adjusting for restricted or biased outcomes.

• Modelling Contexts: Card games, seating plans, and assigning roles.

2.4 Technology Integration

• Calculator Use: Using nCrnCrnCr, nPrnPrnPr, and factorial keys.

• Verification of Answers: Via Casio/TI-Nspire and spreadsheet formulas.