A permutation counts arrangements where order matters, while a combination counts selections where order does not matter. Both methods choose objects from distinct objects, but they count different outcomes.
The Reasoning
For a permutation, changing the order creates a new outcome. For example, assigning Ana as president and Ben as secretary differs from assigning Ben as president and Ana as secretary.
The number of permutations is
For a combination, only the selected group matters. A committee containing Ana and Ben is the same committee as one containing Ben and Ana.
The number of combinations is
The extra division by (r!) removes the (r!) possible internal arrangements of each selected group. Therefore, the two counts are connected by . Every combination of objects can be ordered in (r!) ways, so permutations count each unordered selection (r!) times.
Use a quick decision test: ask whether swapping two chosen objects changes the outcome. If yes, use a permutation; if no, use a combination. Distinct roles and ranked places indicate permutations, whereas unranked teams indicate combinations.
| Feature | Permutation | Combination |
|---|---|---|
| Does order matter? | Yes | No |
| Typical wording | arrange, rank, assign roles, form a code | choose, select, form a committee |
| Formula |
Worked Example
From eight students, three are selected.
If they receive distinct roles of president, secretary and treasurer, order matters:
If they simply form a three-person committee, order does not matter:
A common misconception is that permutations are used whenever objects are placed into groups. The correct test is not whether objects are grouped, but whether rearranging the chosen objects produces a genuinely different outcome.
IB Exam Technique
This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 1.10 Counting principles, and may appear on Paper 1 or Paper 2. Before calculating, explicitly state whether order matters; examiners expect the counting method to follow logically from the context.