Decide whether order changes a count
Choose between a permutation and a combination by asking whether changing the order creates a different outcome.
Section 11 · Permutations and combinations
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the key question when choosing between nPr and nCr?
What you will learn
- Recognise the difference between an arrangement and a selection
- Choose nPr when order matters and nCr when it does not
- Explain the counting choice in a short everyday problem
Ask whether swapping two chosen items changes the result
A permutation counts ordered arrangements. A combination counts unordered selections. If Alex then Bo is different from Bo then Alex, the outcome is ordered; if both names simply belong to the same team, order does not matter.
Roles, ranks, positions and sequences usually make order matter. A committee, group or set of chosen items usually does not. Read the wording to identify what counts as one distinct result before using a formula.
Count one type of outcome at a time
Choosing a captain and deputy assigns two different roles, so there are 7P2 ways to fill them from seven people. Choosing a two-person committee from seven has no roles, so there are 7C2 groups.
For this 0606 topic, use distinct items without repetition. Circular arrangements and problems that combine a selection with a later arrangement are outside the syllabus scope; do not add those extra stages to a count.
Pause and explain
A school assigns gold, silver and bronze to three runners. Does order matter?
Worked example
Seven students are eligible for captain and deputy. How many different assignments are possible?
Show the worked solution
- The captain and deputy roles are different, so swapping two students changes the assignment.
- There are 7 choices for captain, then 6 remaining choices for deputy.
- Multiply the ordered choices: 7 × 6 = 42, which is 7P2.
Answer 42 assignments.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A volunteer group chooses 3 students from 10 for an unranked committee. Should you use a permutation or a combination, and how many committees are possible?
Give me a hint
Changing the order of names on the same committee does not make a new group.
Compare my reasoning
- Order does not matter, so use a combination.
- Calculate 10C3 = 10 × 9 × 8/(3 × 2 × 1).
- Simplify to 120 committees.
Use 10C3; there are 120 committees.
Look for these in your work
- Order test answered
- Combination selected
- Count simplified correctly
A contest has 8 finalists and records first, second and third place. How many different podium results are possible?
Give me a hint
The three finishing positions are distinct, so use an ordered count.
Compare my reasoning
- There are 8 choices for first place.
- Then 7 choices for second and 6 for third.
- Multiply: 8 × 7 × 6 = 336.
8P3 = 336 podium results.
Look for these in your work
- Distinct positions recognised
- Choices decrease without replacement
- Product evaluated
A system creates a four-item sequence from nine distinct symbols, without repeats. If changing position changes the sequence, how many sequences can it create?
Give me a hint
Count the choices for each position in order.
Compare my reasoning
- There are 9 choices for the first position.
- There are then 8, 7 and 6 choices for the remaining positions.
- Multiply: 9 × 8 × 7 × 6 = 3024.
9P4 = 3024 sequences.
Look for these in your work
- Order condition used
- No repetition respected
- Ordered product correct
Common mistakes
- Using a combination when the selected people have different roles
- Using a permutation for a group where no order is recorded
- Counting the same group in every possible order
- Adding a second select-or-arrange stage that the question does not ask for
What can you explain now?
Does choosing three students to form an unranked committee depend on order?
Compare with the explanation
No; use a combination
The same three people form the same committee in every listing order.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
How can you tell from the meaning of an outcome whether order matters?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.