Permutations is one of 2 counting formulas in the probability section of this library, and it is used at high school · ap level.
Why permutations works
Fill the positions one at a time: there are n candidates for the first slot, n minus 1 left for the second, and so on for r slots. Writing that shrinking product as a ratio of factorials is just a compact way of cancelling off the factors you never got around to using.
What each symbol means
$n!$ is factorial.
Permutations: when it holds
$0\le r\le n$ are integers and order matters.
When it stops applying
It assumes all n items are distinguishable and none is reused. The letters of BOOK give 12 distinct arrangements rather than 24, because the two O's are identical, and PIN codes allow repeats so three digits give 1000 options rather than 720.
Permutations: a worked example
Assigning 3 distinct roles among 5 people gives $5!/(5-3)!=60$.
The mistake to avoid
What people do: Counting arrangements when the problem does not care about order.
Why it goes wrong: Choosing 3 people from 5 for distinct roles gives 60 outcomes, but choosing 3 for a committee gives only 10. Using the ordered count multiplies the answer by six here.
Do this instead: Ask whether swapping two of the chosen items produces a different outcome. If it does not, divide by the arrangements of the chosen items, which turns this into a combination.
Permutations: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n!$ is factorial.
- Check the conditions before substituting. $0\le r\le n$ are integers and order matters.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Counting
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where permutations comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about permutations
What does the formula give when r is zero?
It gives 1, since the factorials cancel completely. There is exactly one way to arrange nothing, which sounds odd but keeps every counting formula consistent.
How is this different from a combination?
This counts ordered selections and a combination counts unordered ones. Dividing by the number of ways to arrange the r chosen items converts one into the other.
What about arranging people around a circular table?
Use n minus 1 factorial instead, because rotating everyone one seat around does not create a new arrangement. Fixing one person in place and arranging the rest is the standard trick.
Why is zero factorial defined as 1?
Because it makes these formulas work at the edges. There is one way to arrange an empty set, and defining it any other way would break the pattern that n factorial equals n times the previous factorial.