Mathematics 9709 · syllabus 5.2
Cambridge A Level Mathematics 9709 — Permutations and combinations topical past papers
Counting arrangements and selections: words with repeated letters, people who must or must not stand together, and committees chosen under conditions.
- Questions
- 132
- Years
- 2009–2026
- With mark scheme
- 100%
- Questions 2022–2026
- 42
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What Permutations and combinations covers
This section of Paper 5 (5.2) is about counting. A permutation is an arrangement, where order matters; a combination is a selection, where it does not. You count arrangements of objects in a line using n!, dividing by the factorials of any repeated items, and selections using ⁿCr, then build harder counts by multiplying the choices for successive stages and adding the counts for separate cases.
Restrictions are what make questions hard. Two people who must stand together are treated as one unit and then arranged within it; people who must be apart are handled by subtracting from the total, or by placing everyone else first and slotting them into the gaps. Selections may need at least one of a type, often easiest found by subtracting the unwanted cases from the total, or a split into cases by how many of each type are chosen. Arrangements in a circle are not included.
Counting is also a route to probability in section 5.3: when outcomes are equally likely, a probability is a favourable count divided by a total count, and ⁿCr reappears in the binomial distribution in section 5.4. The ⁿCr values are the same numbers as the coefficients in the binomial expansion of Paper 1. Arrangements in more than one row, such as two rows of seats, are counted the same way as a single line, one position at a time.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 5.2 Permutations and combinations
What you should be able to do
- Count arrangements of objects in a line, including repeated objects
- Count arrangements where items must be together, apart or in fixed positions
- Count selections using ⁿCr, including with conditions on the types chosen
- Split a problem into cases and add, or use the complement and subtract
- Use counts to find probabilities when outcomes are equally likely
Key ideas
- n different objects can be arranged in a line in n! ways
- n objects with p alike of one kind and q alike of another can be arranged in n!/(p! q!) ways
- ⁿPr = n!/(n − r)! ordered arrangements of r from n; ⁿCr = n!/(r!(n − r)!) selections of r from n
- Multiply the counts for successive stages; add the counts for cases that cannot happen together
How Permutations and combinations has been examined
From 2022 to 2026, Permutations and combinations was on Paper 5 · Probability & Statistics 1 in 14 of 14 exam series, worth about 21% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 5 · Probability & Statistics 1 | 42 | 21% | 14 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 10.
Filed in Momentum under: Permutation & Combination.
A worked example
Find the number of different arrangements of the 8 letters of the word BALLOONS, and the number of these in which the two Ls are next to each other.
- BALLOONS has 8 letters, including two Ls and two Os.
- Total arrangements = 8!/(2! × 2!) = 40 320/4 = 10 080.
- For the Ls together, treat LL as a single unit: there are now 7 units, with the two Os still identical.
- Arrangements = 7!/2! = 5040/2 = 2520. The two Ls are identical, so swapping them gives nothing new and there is no extra factor of 2.
Answer: 10 080 arrangements; 2520 with the Ls together
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Using permutations for a committee or a hand of cards, where order does not matter: selections need ⁿCr.
- Treating two people who must be together as one unit but forgetting the 2! ways of arranging them within it.
- Counting 'at least one' cases one by one and counting some of them twice: subtracting the 'none' case from the total avoids this.
- Dividing by the factorial of every repeated letter in the word when only some of the letters are used, or dividing by too few.
Exam technique
- Decide first whether order matters and write it down; then decide whether the problem splits into cases.
- For 'not together', either arrange the others and count the gaps, or subtract 'together' from the total; where time allows, check one method against the other.
- Write each case's count on its own line with its working, then add, since method marks are given case by case.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.