Permutations and combinations
The addition and multiplication rules, permutations and combinations, and how to decide whether order matters. The unit is NF and is the counting base of F5 probability.
What you will learn
- 15.1 Addition rule and multiplication rule.
- 15.2 Permutation: concept and notation (ⁿPᵣ, nPr).
- 15.3 Permutations of distinct objects without repetition, including "three particular objects next to each other". Circular permutation is not required.
- 15.4 Combination: concept and notation (ⁿCᵣ, nCr).
- 15.5 Combinations of distinct objects without repetition.
You need first: Probability
Key terms
| English | 中文 |
|---|---|
| counting principle | 計數原理 |
| addition rule / multiplication rule | 加法原理／乘法原理 |
| permutation | 排列 |
| combination | 組合 |
| factorial | 階乘 |
| arrangement, selection | 排列方法，選取方法 |
| at least, at most | 最少，最多 |
Standard steps
The first question: does order matter?
| Order matters (arrange: queue, password, seats, positions) | Order doesn't matter (choose: team, committee, hand of cards) |
|---|---|
| permutation ⁿPᵣ = n!/(n − r)! | combination ⁿCᵣ = n!/[r!(n − r)!] |
- Ask: if two of the chosen items swap places, is it a different outcome? If yes, order matters.
- Order matters: use ⁿPᵣ (n! to arrange all n). Order doesn't matter: use ⁿCᵣ.
- Check the size: ⁿPᵣ = ⁿCᵣ × r!, so the permutation count is always the larger one.
"And" multiplies, "or" adds
- Choose 3 boys and 2 girls: multiply.
- Case 1 or case 2: add the cases.
Standard patterns
| Situation | Method |
|---|---|
| A and B must be together | glue them into one block; arrange the blocks; × 2! for the order inside |
| A and B must not be together | total − together |
| at least one … | total − none |
| particular person must be included | put them in; choose the rest from the others |
| first and last positions fixed | fill those first, then the middle |
Worked examples
Example 1: How many ways to arrange the letters of MATHS?
5! = 120.
Example 2: Five people stand in a row. A and B must be next to each other.
glue AB: 4 blocks → 4! = 24
A and B can swap: × 2! = 2
24 × 2 = 48Example 3: Choose a committee of 3 boys from 6 and 2 girls from 5.
⁶C₃ × ⁵C₂ = 20 × 10 = 200.
Example 4: Choose 4 people from 6 boys and 4 girls, with at least one girl.
total − no girls = ¹⁰C₄ − ⁶C₄ = 210 − 15 = 195Common mistakes
- Using P when the order doesn't matter (or C when it does).
- Forgetting × 2! for the order inside a glued block.
- "At least one" done case by case, missing or double-counting a case.
- Adding when the question needs "and" (multiply).
- Counting circular arrangements (not required, but students sometimes try).
How the DSE tests it
- Section B probability questions use ⁿCᵣ.
- Paper 2 Section B: counting arrangements with conditions (together, not together, at least one).
Questions you might not ask in class
How do I tell P from C?
Ask: if I swap two people, is it a different result? Different seats: yes, P. Same team: no, C.
Why is 0! = 1?
There is exactly one way to arrange nothing: do nothing. It also makes the formulas work.
Why is 'total minus none' easier for 'at least one'?
"At least one" has many cases (1, 2, 3, …). "None" is one case. Subtracting it from the total is quicker and safer.