Permutations and combinations are counting methods that help you find how many outcomes are possible without listing every case. They are useful in probability, games, passwords, schedules, genetics, and many other situations where choices can be arranged or grouped. The key question is whether order matters, because arranging the same items in a different order may or may not create a new outcome.
Factorials give a compact way to count long chains of choices.
Key Facts
- Factorial: n! = n(n - 1)(n - 2)...(1), and 0! = 1
- Permutation of n items taken r at a time: P(n, r) = n!/(n - r)!
- Combination of n items taken r at a time: C(n, r) = n!/(r!(n - r)!)
- Use permutations when order matters, such as 1st, 2nd, and 3rd place winners.
- Use combinations when order does not matter, such as choosing a team or committee.
- Connection: P(n, r) = C(n, r) r!, because each group of r items can be arranged in r! orders.
Vocabulary
- Permutation
- A selection and arrangement of items where order matters.
- Combination
- A selection of items where order does not matter.
- Factorial
- The product of a positive integer and all smaller positive integers, with 0! defined as 1.
- Sample space
- The set of all possible outcomes in a counting or probability problem.
- Repetition
- A rule that tells whether an item can be chosen more than once.
Common Mistakes to Avoid
- Using a permutation when order does not matter. This overcounts because the same group is counted once for every possible ordering.
- Using a combination when order matters. This undercounts because different arrangements, such as ABC and BAC, should be treated as different outcomes.
- Forgetting the factorial in the denominator of C(n, r). The r! removes duplicate orderings of the same selected group.
- Assuming repetition is allowed without checking the problem. The formulas P(n, r) = n!/(n - r)! and C(n, r) = n!/(r!(n - r)!) apply when items are chosen without replacement.
Practice Questions
- 1 A race has 8 runners. How many ways can gold, silver, and bronze medals be awarded if no runner can win more than one medal?
- 2 A club has 12 members. How many different 4 person committees can be chosen?
- 3 A teacher chooses 3 students from a class to stand in a line for a demonstration. Explain whether this situation should be counted with a permutation or a combination, and why.