Permutations and combinations
By the end, you can
- Decide whether order matters
- Count arrangements with restrictions or repetitions
- Use addition and multiplication without double counting
about 30 minutes for the lesson · practise at your own pace.
The central idea
Use addition for disjoint alternatives and multiplication for successive choices. Selecting r objects from n distinct objects without order gives . Assigning ordered roles gives . Repeated identical objects reduce arrangement counts; circular arrangements require an explicit convention about rotations.
Worked example
Choose two representatives from five people: . Choose a chair and a secretary instead: 5 choices then 4 choices, giving 20. Swapping roles changes the outcome in the second problem.
Multiplication counts sequences of choices; it may count the same final unordered set more than once.
Check your understanding
How many distinct arrangements are there of the letters in LEVEL?
There are five letters, with two Ls and two Es, so .
Mastery learning objectives
At the end of this chapter, I should be able to:
Choose a practice question
Use the printed question-type map to choose a method to practise. Links open the complete question; the printed part references are retained below.
| Skill | Lecture examples | Discussion | Review follow-up |
|---|---|---|---|
| Distinct-object permutations with restrictions | 3, 4, 5, 13, 15(i), 16 | 1, 2, 13(i) | 9(i), 13, 15(i)(i), 16(last part) |
| Repeated-object permutations with restrictions | 6, 12, 14(a), 15(ii) | 4, 5, 7, 10 | 2, 3, 5(a), 7(a), 8, 10, 11, 12(b), 14 |
| Selection, including restrictions | 7, 8, 9(i)(ii) | 12(b) | 1 |
| Selection into non-distinct groups of the same size | 10 | 3 | — |
| Circular arrangements | 18, 19 | 8 | 5(b), 6, 7(b), 9(ii), 12, 14(iii), 15(iii)(iv), 16(i)(ii) |
| Complement | 8(iv), 9(iii) | 4(iii) | 2(iii) |
| Grouping and separation | 13, 16(iii) | 5(ii), 6, 11(a) | 4 |
| Cases | 8(ii)(iii), 11, 14(b), 15(iii), 16(ii), 17 | 5(iv), 7(ii), 9, 10(ii), 11(b), 13(ii) | 7(a), 10(b)(i) |
Read the printed references critically. The complement row prints Example8(iv), whose source working uses inclusion–exclusion and disjoint cases. The explicit complement method is in Example8(iii), “at least one girl”. Both are available in the linked Example8.
The printed Review15(i)(i) repeats(i); the link opens Review15 without inventing a missing part.
The repeated-object row includes Example15(ii), which actually allows repetition of the available distinct digits.
Preserve these source references as navigation pointers while deciding the counting model from each question. Some questions use more than one strategy.
Practice questions
55 examples and practice questions, in 10 sections.
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