# Permutations and Combinations

> CIE IGCSE Additional Mathematics · CIE IGCSE Add Maths 0606
> Source: https://www.owlsprep.com/study/cie-0606-u11-overview/
> Weight: 5-8% of Paper 1 and Paper 2 combined

This unit introduces core counting principles for permutations and combinations, essential for solving the arrangement and selection problems tested in CIE IGCSE Additional Mathematics exams.

**Prerequisites:** Understanding of factorial notation ($n!$)

## Learning objectives

- Distinguish between permutations and combinations based on whether order of selection matters
- Calculate permutations of n distinct items taken r at a time, including problems with restrictions like adjacent or excluded items
- Calculate combinations of n distinct items taken r at a time, including grouped and conditional selection problems
- Apply permutation and combination rules to solve real-world and exam-style counting problems

## 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](https://www.owlsprep.com/study/cie-0606-u11-permutations-and-combinations/) — Covers definitions, formulas, restriction rules, and step-by-step guidance for solving common exam-style counting problems.

## Common pitfalls

- **Wrong:** Using permutations for unordered selection problems
  - Why it fails: Permutations count ordered arrangements, so using them for unselected groups overcounts identical sets of items
  - Correct: Use combinations for unordered groups, permutations only when sequence or position matters
- **Wrong:** Treating 'must be together' items as a single block without arranging within the block
  - Why it fails: Items required to be adjacent form one unit, but the items inside that unit can still be arranged among themselves
  - Correct: Bundle the adjacent items as one unit, arrange the units, then multiply by the internal arrangements of the bundled items
- **Wrong:** Double-counting combinations with overlapping selection criteria
  - Why it fails: Counting the same group multiple times when applying layered selection rules leads to overestimates
  - Correct: Use complementary counting or partition groups into non-overlapping sets to avoid duplicates

## Cheatsheet

| Formula | Use Case |
| --- | --- |
| $n! = n \times (n-1) \times ... \times 1$ | Calculate factorial of n, used in all permutation/combination calculations |
| $^nP_r = \frac{n!}{(n-r)!}$ | Number of ordered permutations of r items selected from n distinct items |
| $^nC_r = \frac{n!}{r!(n-r)!}$ | Number of unordered combinations of r items selected from n distinct items |
| $^nP_r = r! \times {}^nC_r$ | 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 $^nC_r$ coefficients are used to build binomial expansions.

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