Before you begin
What you need: Whole-number multiplication and factorial notation.
- Use the multiplication principle
- Distinguish permutations from combinations
- Handle simple restrictions and repeated objects
Understand the idea
Count choices in stages
If a task has a choices at the first stage and b choices at the second for every first-stage choice, there are ab outcomes. For example, 3 shirts and 2 trousers make 6 outfits.
Arrangements care about order
Selecting a captain then a vice-captain creates distinct ordered roles. From n distinct people without repetition, the choices are n, then n − 1, and so on. Their product is a permutation.
Selections ignore order
A committee does not distinguish selection order. Each group of r distinct people appears r! times among ordered arrangements, so divide by r!. Factorial n! multiplies integers from 1 to n; 0! = 1.
Choosing two from A, B and C
| Selection | Its ordered arrangements |
|---|---|
| A and B | AB, BA |
| A and C | AC, CA |
| B and C | BC, CB |
A method you can reuse
nPr = n!/(n − r)!; nCr = n!/[r!(n − r)!]
For integers 0 ≤ r ≤ n, distinct objects, and selection without repetition.- Decide whether order or assigned roles make outcomes different.
- Check whether repetition is allowed and whether objects are distinct.
- Count the valid stages, then divide out duplicates only when justified.
nCr = nC(n − r): choosing who joins a group determines exactly who stays out.
Worked examples
Read the question first. Try a step yourself, then compare your reasoning.
Arrange books
How many ways can 4 distinct books be arranged in a row?
- There are 4 choices for the first position.
- Then 3, 2 and 1 choices remain.
- Multiply 4 × 3 × 2 × 1.
Choose a committee
Choose 2 people from 5 for a committee with no distinct roles.
- Order does not matter.
- 5C2 = (5 × 4)/(2 × 1).
- Divide by 2! because each pair is otherwise counted twice.
Repeated letters
How many distinct arrangements of the letters in MOM?
- Treating all three positions as distinct gives 3! = 6.
- The two M letters are identical; swapping them changes nothing.
- Divide by 2!.
Check your understanding
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