Permutations and Combinations

Permutations and Combinations is the middle link of the likelihood chain on Paper Two: it builds on the combinations concept from Binomial, and it leads directly into Probability. It is the counting toolkit — and counting outcomes is what probability questions spend most of their marks on.

Like Binomial, this topic does not headline the marks tables: its questions are generally embedded within probability questions, so the marks accrue under the probability label. But a probability question is usually two counting problems in disguise — the favourable outcomes and the total outcomes — and this topic is where you learn to count them.

The topic starts with counting principles: how to count the total outcomes when you track several events together, like a die roll and a coin flip at once.

Then the permutations half: the concept of sequential selection — order matters, and once something is selected it cannot be selected again — followed by the factorial notation that explains where the permutation formula comes from, and multi-stage permutation problems that chain several arrangements into one count.

The combinations half mirrors it: the logic behind the combination formula in factorials (building on the concept you met in Binomial), then multi-stage combination problems — choosing this many from here and that many from there in one question.

The topic closes with "at least one" problems, solved by complement counting: instead of counting the scenarios with at least one of something, count the scenarios with zero of it and subtract from the total.

Like Binomial, this topic has no correlations table of its own: in the past-paper data, permutations and combinations questions are generally embedded within probability questions, so all the correlation signal accrues to probability instead.

The categorisation is telling you how the topic is examined. A probability question hands you a scenario — cards, teams, dice — and the first thing it silently demands is a count: how many ways can this happen, out of how many ways in total? That demand is this topic. The better you count, the more of the probability marks open up.

The chain here is short and clean: Binomial feeds in, and Probability is where it all goes.

There are seven sub-topics in Permutations and Combinations: the counting principles first, then the permutations half, then the combinations half, and one closing technique. Click each step to see how they build.