Explore six lessons of puzzles, patterns and powerful counting ideas.

Counting & Combinatorics begins with a simple question: how can we count every possibility without missing one or counting anything twice? Across six lessons, learners progress from systematic lists and tree diagrams to multiplication, arrangements, combinations, Pascal’s triangle and flexible counting strategies.

The course is designed for exploration rather than memorising formulas. Learners are encouraged to organise possibilities, test ideas, compare methods and explain why their counting works.

The complete course includes

Lesson 1: How many possibilities? — systematic lists, tables and tree diagrams;
Lesson 2: Choices in stages — the multiplication principle and multi-stage counting;
Lesson 3: Arrangements — order, factorials and restricted arrangements;
Lesson 4: Choosing without ordering — combinations, teams and selections;
Lesson 5: Pascal’s triangle — combination numbers, symmetry and patterns;
Lesson 6: Clever counting — complements, cases, symmetry, double counting and other strategies;
printable handouts;
supplementary exercises;
answers and discussion.

Suitable for learners aged 10 and over. No formal prerequisites are required, although the lessons are designed to be followed in order.

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This download brings together the full 6-part Beyond Curriculum: Counting & Combinatorics course.

Pascal’s triangle

Counting & Combinatorics – Lesson 5: Pascal’s Triangle

Counting & Combinatorics — Complete Course