Permutations
Number of ordered arrangements of r items from n.
Formula
Variables
Example
P(5,2) = 5!/3! = 20
How it works
Permutations count how many ways you can choose and arrange r items from a pool of n when order matters. The formula works by taking n choices for the first position, then n-1 for the second, and so on until r positions are filled; multiplying these gives the same result as n!/(n-r)! because the unused items cancel out. If changing the order makes a different outcome, this is the right formula to use.
When to use it
Use this for arranging people, objects, or labels in a line, assigning ranked positions, or selecting ordered codes/password-like sequences without repetition. It is especially useful in probability and counting problems where the sequence of selection changes the result.
Worked example, step by step
Find the number of ordered arrangements of 3 students chosen from 7. P(7,3) = 7! / (7-3)! P(7,3) = 7! / 4! P(7,3) = (7 × 6 × 5 × 4!) / 4! P(7,3) = 7 × 6 × 5 P(7,3) = 210 So, there are 210 possible ordered arrangements.
Common mistakes to avoid
- Using permutations when the order does not matter
- Forgetting to subtract r from n in the denominator
- Including repeated choices when the problem says items cannot be reused
Did You Know?
Order matters in permutations, unlike combinations.
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More in Probability & Combinatorics
View allCombinations
BasicNumber of unordered selections of r items from n.
Factorial
BasicProduct of all positive integers up to n.
Permutations with Repetition
BasicOrdered arrangements of r items from n when repetition is allowed.
Permutations of a Multiset
IntermediateArrangements of n items where some are identical.
Classical Probability
BasicProbability of an event with equally likely outcomes.
Complement Rule
BasicProbability that an event does not occur.