# Permutations and combinations

A permutation counts arrangements where order matters: nPr = n! / (n - r)!. A combination counts selections where order does not matter: nCr = n! / (r! (n - r)!). The symbol n! means n x (n-1) x ... x 1.

Statistics (university) · Part 02: Probability · Card 8 of 8 · Last updated 2026-10-19

## Analogy

Choosing a president, secretary and treasurer is a permutation. Choosing three people for a committee is a combination.

## Example

**Pick 3 from 5**

Ordered: 5 x 4 x 3 = 60
Unordered: 60 / 3! = 60 / 6 = 10
So 5P3 = 60
and 5C3 = 10

## In practice

Combinations count lottery tickets and card hands. Permutations count race podium orders.

## Practice

### Flashcards

- **Value of 5C2?** 10. That is 5 x 4 / 2.

### Interview questions

**A club has 8 members. How many ways can it pick a committee of 3, and how many ways can it assign president, secretary and treasurer? Explain the difference.**

- Committee: 8C3 = 56
- Officers: 8P3 = 336
- Order matters for officers
- 336 / 6 = 56

## Questions people ask

### How do I decide between a permutation and a combination?

Ask whether swapping two chosen items gives a different result. If yes, order matters, so use a permutation. If no, use a combination. A podium of gold, silver and bronze is ordered. A group of three friends is not, because the same three friends are the same group.

### Why divide by r factorial in the combination formula?

A permutation counts each group once for every way of ordering it. A group of r items can be ordered in r! ways. Dividing by r! removes those repeats. For 3 items that is 6 orders, so 60 permutations become 10 groups.

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