Direct answer
Use permutations when arrangement order matters and combinations when only the selected group matters; then decide separately whether repetition is allowed.
What this calculation tells you
Permutations count ordered arrangements; combinations count unordered selections. Both arise from the multiplication principle and factorial structure.
The labels attached to objects and the equivalence rule determine whether swapping two selections creates a new outcome.
Where it is used
Scheduling and assignment
Count ordered positions, roles, or sequences.
Sampling and selection
Count groups where internal order is irrelevant.
Computing and security
Estimate bounded strings or credential spaces under explicit character rules.
Probability
Build equally likely outcome counts only when the sampling model supports them.
When this guide helps
- Choosing a committee.
- Assigning first, second, and third places.
- Selecting with or without replacement.
- Accounting for repeated symbols.
Ask whether swapping changes the outcome
If AB and BA are distinct, order matters. If they represent the same selected group, use an unordered model.
Decide whether objects can repeat
Replacement, reusable character pools, and repeated object types change the count and must be specified separately from order.
Define identical outcomes
Circular arrangements, repeated letters, and indistinguishable objects require equivalence adjustments rather than the simplest nPr or nCr formula.
Common mistakes
Typical errors include choosing a formula by vocabulary alone and treating repeated symbols as distinct.
- Describe one outcome.
- Test a swap.
- State repetition and identity rules.
Worked case: order matters
Choose a president and vice-president from five people.
There are 5 choices for president and 4 remaining for vice-president: 5x4=20.
There are 20 ordered assignments.
Swapping the two people creates a different office assignment, so permutation logic applies.
Reproduce this worked caseOpen Combination Calculator
Worked case: order does not matter
Choose a two-person committee from the same five people.
There are 5C2=5!/(2!3!)=10 unordered pairs.
There are 10 committees.
Each pair would be counted twice by 5P2 because AB and BA describe the same committee.
Reproduce this worked caseOpen Combination Calculator
Compare permutations and combinations cases before generalising
Replacement, repetition and distinguishability can change the model. Define what counts as a distinct outcome before applying a formula.
| Task | Order matters? | Count |
|---|---|---|
| President and vice-president | Yes | 5P2=20 |
| Two-person committee | No | 5C2=10 |
| Choose 3 from 10 | No | 10C3=120 |
permutations and combinations: calculation checklist
- Define distinct outcome
- Decide whether order matters
- Decide replacement/repetition
- Require 0≤r≤n for basic model
- Check small cases by listing
Practical questions
Frequently asked questions
Is a lottery a combination?
A draw is combination-like only when ticket order does not matter under the stated game rules.
Why are permutations usually larger?
Each selected group can often be arranged in several distinct orders.
What if objects repeat?
Use a repetition or multiset model matching whether positions and copies are distinguishable.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.