How many possibilities are there?
How many different outfits can you make? How many codes are possible? How many ways can objects be arranged? Combinatorics turns questions like these into mathematics by finding clever ways to organise possibilities so that nothing is missed and nothing is counted twice.
The six lessons
How many possibilities?
Explore systematic ways to count possibilities using lists, tables and tree diagrams.
Choices in stages
Discover why choices made in stages can often be counted by multiplying.
Arrangements
Learn how to count different orders and meet factorials along the way.
Choosing without ordering
See what changes when the group matters but the order does not.
Pascal’s triangle
Explore the patterns hidden in Pascal’s triangle and their connection to combinations.
Clever counting
Bring the ideas together using cases, complements, symmetry and double counting.
How to use this course
There’s no need to rush. This course is designed to be explored by thinking, drawing, experimenting and trying things out.
When you reach a problem, have a go before opening the answer — even if you’re not sure where to start. Getting stuck for a while can be useful.
Look out for the 💡 icons. Click one if you’d like a suggestion about how to approach a problem or get more from an idea. They’re completely optional, so confident learners can simply ignore them.
When you find a way to count something, ask yourself: How do I know I haven’t missed anything? and Have I counted anything twice?
Sometimes two approaches will both work. Try to notice which one makes the structure of the problem easiest to see.
And don’t feel you have to complete everything in one sitting. Stopping with an interesting question still in your head is a perfectly good place to finish.
Download the complete six-lesson collection of printable handouts, exercises and answers in one place.
View the complete course resources →