Permutations
Permutations count ordered arrangements.
About this topic
A permutation is an arrangement of objects where order matters. The number of ways to arrange n distinct objects is n! (n factorial) = n × (n − 1) × … × 1. Arranging r objects chosen from n gives nPr = n!/(n − r)!.
Repeated items reduce the count: the letters in MISSISSIPPI can be arranged in 11!/(4! 4! 2!) = 34,650 ways, dividing by the factorials of the repeated letters. Circular arrangements of n objects number (n − 1)!, because rotations look the same. By definition 0! = 1.
Example: in a race with 8 runners, the number of ways to award gold, silver and bronze is 8P3 = 8 × 7 × 6 = 336. Arranging 5 books on a shelf can be done in 5! = 120 ways. A locker combination where order matters, such as 3 distinct numbers from 40, has 40 × 39 × 38 = 59,280 arrangements.
Permutations are used for scheduling, ranking, cryptography and puzzle solving such as the Rubik's cube, whose count of arrangements exceeds 43 quintillion. To decide whether a problem is a permutation, ask: would swapping two items give a different result? If so, order matters.
Key ideas
- nPr = n!/(n − r)!
- Order matters
- 5 people in a line: 5! = 120
Quick check
Use cases: where Permutations helps
- Homework: review the key idea behind Permutations before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Permutations without notes.
- Visual learning: play with the interactive lab to see how Permutations changes when you adjust its inputs.
- Teaching a friend: explain Permutations in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Permutations?
Permutations count ordered arrangements.
What will I learn in Permutations?
nPr = n!/(n − r)! Order matters. 5 people in a line: 5! = 120.
Is Permutations part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Permutations lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Permutations?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.