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.
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Analogy
Choosing a president, secretary and treasurer is a permutation. Choosing three people for a committee is a combination.
Example
Ordered: 5 x 4 x 3 = 60Unordered: 60 / 3! = 60 / 6 = 10So 5P3 = 60and 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.