Permutations or combinations?
The hardest part is not evaluating a formula. It is deciding whether swapping two selected objects creates a new outcome.
The swap test
podium · order matters
A → B ≠ B → A
permutation
team · order ignored
{A, B} = {B, A}
combination
permutation ÷ r! = combination
Ask whether a swap changes the outcome
- Different outcome after a swap: order matters, so use a permutation.
- Same outcome after a swap: order is ignored, so use a combination.
- Named roles such as captain and vice-captain make order matter even when the wording says “choose”.
Definition
1 markWhat is meant by a permutation?
Model answer: A selection and arrangement of objects in which order matters.
Definition
1 markWhat is meant by a combination?
Model answer: A selection of objects in which order does not matter.
A combination removes the internal order
Symbols
- = number available (objects)
- = number selected (objects)
- = unordered selections (ways)
A permutation counts every internal order of the chosen group. Divide by those orders to get .
Choosing the objects to include also determines the objects left out, so . This is a useful calculator check.
Worked example
Same pool, different question
- Arrange 3 of 7 people into named roles: .
- Choose a three-person team from 7 people: .
Each team has internal orders, so .
Examiner note
Choose the model before nPr or nCr
- Words such as “arrange” and “select” are clues, not complete rules. Use the swap test on the actual outcome.
- Use nPr for ordered positions or roles without reuse. Use nCr for an unordered group.
- The calculator is the fastest way to evaluate either expression. Write the expression so the method is still visible.
Common mistake
From 8 students, find the number of ways to choose (a) two class monitors with the same role and (b) a captain and a vice-captain.
Show worked answer
Swapping the monitors changes nothing; swapping the named roles does.