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.

In Lesson 6 of Counting & Combinatorics, learners bring together the ideas from the course and explore a collection of powerful counting strategies.

They investigate counting the opposite, splitting a problem into cases, using symmetry, counting the same thing in two ways and replacing a complicated object with a simpler one.

This lesson includes

• complementary counting — counting what is easier instead;
• splitting problems into separate cases;
• using symmetry to avoid repeated work;
• double counting and counting the same set in two ways;
• matching complicated objects with simpler objects;
• problems involving coin tosses, codes, handshakes, diagonals and subsets;
• a rectangle-counting challenge;
• answers and discussion;
• supplementary exercises for extra practice and revision.

Suitable for learners aged 10 and over. This lesson draws on ideas developed throughout Lessons 1–5.

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This is Lesson 6 of the 6-part Beyond Curriculum: Counting & Combinatorics course.

In Lesson 5 of Counting & Combinatorics, learners build Pascal’s triangle row by row and investigate the patterns hidden inside it. They then connect each row with the numbers of ways to choose different-sized groups.

The lesson explains why neighbouring entries add together, explores symmetry and row sums, and shows how Pascal’s triangle can answer combination questions without starting again from scratch.

This lesson includes

• building and continuing Pascal’s triangle;
• the rule that each inside entry is the sum of the two above it;
• the connection between Pascal’s triangle and combinations;
• committee and team-selection problems;
• an explanation of the Pascal recurrence using two cases;
• symmetry and other patterns in the triangle;
• answers and discussion;
• supplementary exercises for extra practice and revision.

Suitable for learners aged 10 and over. Familiarity with combinations from Lesson 4 is recommended.

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This resource is completely free. If you find it useful and would like to
support the creation of more free mathematics resources, you can

support Sumthing Else on Ko-fi
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This is Lesson 5 of the 6-part Beyond Curriculum: Counting & Combinatorics course.

In Lesson 4 of Counting & Combinatorics, learners compare arrangements with selections such as teams, committees and pizza toppings. They see why Alice–Ben and Ben–Alice describe the same team and why ordered counting can overcount.

The lesson develops from choosing two people to choosing larger groups, leading to combination notation and the idea of dividing by the orders that do not matter.

This lesson includes

• practice deciding whether order matters;
• two-person team examples showing duplicate ordered choices;
• choosing larger groups by arranging first and dividing by unwanted orders;
• combination notation;
• the symmetry between choosing what stays and choosing what is left out;
• team, committee and pizza-topping problems;
• answers and discussion;
• supplementary exercises for extra practice and revision.

Suitable for learners aged 10 and over. Lesson 3 on arrangements is recommended first.

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This resource is completely free. If you find it useful and would like to support the creation of more free mathematics resources, you can support Sumthing Else on Ko-fi.

This is Lesson 4 of the 6-part Beyond Curriculum: Counting & Combinatorics course.

In Lesson 3 of Counting & Combinatorics, learners explore arrangements by placing books, people and letters in different orders. They discover how the multiplication principle leads naturally to factorials.

The lesson then develops the idea through fixed positions, restricted positions and problems where two objects must stay together.

This lesson includes

• a clear introduction to arrangements and why order matters;
• worked examples using books, people and letters;
• factorial notation and the rule for arranging n different objects;
• problems with fixed and restricted positions;
• the block method for objects that must stay together;
• a challenge on keeping two friends together;
• answers and discussion;
• supplementary exercises for extra practice and revision.

Suitable for learners aged 10 and over. Familiarity with the multiplication principle from Lesson 2 is helpful.

☕ Free to download

This resource is completely free. If you find it useful and would like to
support the creation of more free mathematics resources, you can

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This is Lesson 3 of the 6-part Beyond Curriculum: Counting & Combinatorics course.

In Lesson 2 of Counting & Combinatorics, learners discover the multiplication principle: when a process happens in stages, the number of possibilities can often be found by multiplying the number of choices available at each stage.

The lesson develops this idea through outfits, ice creams, game characters, codes, café orders and coloured-ball problems. Learners also explore what happens when earlier choices reduce the number of later options, and when separate cases should be added rather than multiplied.

Included

• Lesson Handout — guided examples, practice, learning prompts and the Mystery Balls challenge.
• Exercises — extra problems on multi-stage counting, no repetition, separate cases and working backwards from a total.
• Answers and discussion at the back of both resources.

Topics include:

• the multiplication principle;
• two-stage and multi-stage choices;
• counting codes;
• choices without repetition;
• multiplying through stages;
• adding separate cases;
• factor problems and working backwards.

The materials are designed for a parent and child to explore together or for an older learner to work through independently.

This is the printable companion to Lesson 2: Choices in stages in the free Sumthing Else Counting & Combinatorics course.

☕ Free to download

This resource is completely free. If you find it useful and would like to
support the creation of more free mathematics resources, you can

support Sumthing Else on Ko-fi
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In Lesson 1 of Counting & Combinatorics, learners explore systematic ways of organising possibilities using lists, tables and tree diagrams.

Along the way they investigate outfits, codes, handshakes and triangles, and begin thinking about an important question in combinatorics: does order matter?

Included

• Lesson Handout — guided examples, practice, learning prompts and a challenge.
• Exercises — extra questions for practice, extension and revision.
• Answers and discussion at the back of both resources.

The materials are designed for a parent and child to explore together or for an older learner to work through independently.

Topics include:

• systematic lists;
• tables;
• tree diagrams;
• counting codes;
• avoiding double counting;
• deciding whether order matters.

This is a printable companion to Lesson 1: How many possibilities? in the free Sumthing Else Counting & Combinatorics course.

☕ Free to download

This resource is completely free. If you find it useful and would like to
support the creation of more free mathematics resources, you can

support Sumthing Else on Ko-fi
.