Unit Overview

Permutations and Combinations

CIE IGCSE Additional Mathematics· 5 min read 📊 5-8% of Paper 1 and Paper 2 combined

1. Unit at a Glance

Counting problems make up a consistent portion of your exam marks, and this unit builds a logical framework for deciding which counting rule to use for any given problem. You will move from basic definitions to applying rules to complex problems with constraints, such as mandatory adjacent items or restricted selection groups.

This unit contains one core sub-topic covering all required content for permutations and combinations:

Permutations and Combinations

Covers definitions, formulas, restriction rules, and step-by-step guidance for solving common exam-style counting problems.

★★★⏱ 15 min

2. Common Pitfalls

Wrong move:

Using permutations for unordered selection problems

Why:

Permutations count ordered arrangements, so using them for unselected groups overcounts identical sets of items

Correct move:

Use combinations for unordered groups, permutations only when sequence or position matters

Wrong move:

Treating 'must be together' items as a single block without arranging within the block

Why:

Items required to be adjacent form one unit, but the items inside that unit can still be arranged among themselves

Correct move:

Bundle the adjacent items as one unit, arrange the units, then multiply by the internal arrangements of the bundled items

Wrong move:

Double-counting combinations with overlapping selection criteria

Why:

Counting the same group multiple times when applying layered selection rules leads to overestimates

Correct move:

Use complementary counting or partition groups into non-overlapping sets to avoid duplicates

3. Quick Reference Cheatsheet

Formula	Use Case
$n! = n \times (n - 1) \times ... \times 1$	Calculate factorial of n, used in all permutation/combination calculations
$^{n} P_{r} = \frac{n !}{( n - r )!}$	Number of ordered permutations of r items selected from n distinct items
$^{n} C_{r} = \frac{n !}{r ! ( n - r )!}$	Number of unordered combinations of r items selected from n distinct items
$^{n} P_{r} = r! \times^{n} C_{r}$	Links permutations and combinations; useful for algebraic 'solve for n' problems
Total arrangements - invalid arrangements	Complementary counting for problems with multiple constraints

What's Next

Begin by working through the core Permutations and Combinations sub-topic linked below, where you will learn definitions, rule derivations, and worked examples for all common exam problem types. Once you have mastered these counting rules, move on to the Series unit, where the same $^{n} C_{r}$ coefficients are used to build binomial expansions.