What you will learn in Permutations, Combinations & the Binomial Theorem

1. 1. Addition and Multiplication Principles

   Two rules that between them handle every counting problem. Almost every mistake is using one where the other belongs.

2. 2. Factorials and Arrangement Counts

   n! is not a definition to memorise. It is what the multiplication principle gives when every stage uses up one option.

3. 3. Permutations

   Ordered selections. The first of four models, and the classifier that tells them apart.

4. 4. Permutations with Restrictions

   Three techniques — glue, complement, divide by repeats — cover almost every constraint you will meet.

5. 5. Combinations

   When order stops mattering, divide by the number of orders. Shown as a grouping, not asserted as a rule.

6. 6. Selection Problems and Partitions

   Committees, hands and routes — and the double-counts that make plausible arguments wrong.

7. 7. Pascal's Triangle and Combinatorial Identities

   Every pattern in the triangle is a counting argument. The recurrence itself is one.

8. 8. The Binomial Theorem

   The coefficients are combination counts because expanding is choosing. That is why this topic sits with counting.

9. 9. General Term, Middle Term, Specific Coefficient

   Finding one term without expanding. Solve for r rather than multiplying everything out.

10. 10. Approximation and Divisibility with Binomials

    “For small x” — how small? The answer is a number, and it moves when your assumptions move.