5.2 Permutations and combinations
- Syllabus
- 9709–2028–2029
- Topic
- 5.2
- Level
- AS
Learning objectives
5.2.1Permutations and combinations• understand the terms permutation and combination, and solve simple problems involving selections5.2.2Arrangement restrictions• solve problems about arrangements of objects in a line, including those involving - repetition (e.g. the number of ways of arranging the letters of the word 'NEEDLESS') - restriction (e.g. the number of ways several people can stand in a line if two particular people must, or must not, stand next to each other). Questions may include cases such as people sitting in two (or more) rows. Questions about objects arranged in a circle will not be included.
Permutations count ordered selections while combinations ignore order
nPr counts ordered arrangements of r objects from n; nCr counts unordered selections. Use nPr=n!/(n−r)! and nCr=n!/[r!(n−r)!].
Decide whether positions matter before choosing a formula, and avoid counting the same outcome under different descriptions.
Choosing president and secretary from 8 people uses 8P2; choosing a two-person committee uses 8C2.
The same two people can form two ordered roles but only one unordered committee.
Choose division, blocks, complements or row allocation
| Structure | Counting method |
|---|---|
| repeated identical objects | n!/(a!b!⋯) |
| specified objects together | treat as one block, then multiply internal orders |
| specified objects not together | total arrangements − together arrangements |
| people in two or more rows | choose/allocate people to labelled seats/rows, then arrange within rows |
NEEDLESS has $8$ letters with $E$ repeated $3$ times and $S$ repeated $2$ times, so\frac{8!}{3!2!}distinctlineararrangements.
For 5 distinct books with two specified together: 4!×2!. For them not together: 5!−4!×2!. State whether ends/particular seats create separate cases.
If rows or seats are distinguishable, allocate occupants with combinations or direct seat choices, then arrange each row. Avoid counting the same final seating through different allocation orders.
Circular arrangements are explicitly excluded. Do not identify rotations or use (n−1)!; every task here is a line or labelled-row arrangement.