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.
Common situations
- 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.
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.