Free Permutation and Combination Calculator
Count ordered permutations P(n,r) and unordered combinations C(n,r).
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Inputs
Result
P(10, 3) — permutations
720
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Short answer
P(n, r) = n! / (n − r)! counts ordered arrangements. C(n, r) = n! / (r! · (n − r)!) counts unordered selections.
What is the Permutation and Combination Calculator?
Two ways to count subsets of size r from n items: with order (permutations) or without order (combinations).
How does the Permutation and Combination Calculator work?
Compute factorials in the numerator and denominator; the ratio simplifies to a compact integer.
Formula
Variables
n!n · (n−1) · … · 2 · 1rItems selected from n
Explanation
Combinations discount permutations by r! — the number of orderings within any single selected group.
Examples
Example 1: n = 10, r = 3
P(10,3) = 720 arrangements; C(10,3) = 120 groups.
Applications
- Lottery odds
- Poker probabilities
- Assigning committees or seats
Advantages
- Handles n up to 170 (double-precision factorial limit)
Limitations
- Assumes distinguishable items and no repetition — use multinomials otherwise
Common mistakes
- Using permutations when order doesn't matter (or vice versa)
Tips
- Ask: does swapping the order of two chosen items give a different outcome? If yes → permutation.
Related concepts
The Permutation and Combination Calculator sits inside the Math Calculators hub, in the statistics & probability cluster. Describing data and quantifying uncertainty. Understanding the terms below makes the output easier to interpret and easier to compare against neighbouring measures.
Practical use cases and industry applications
Understanding Permutation and Combination
Two ways to count subsets of size r from n items: with order (permutations) or without order (combinations). Within math calculators, permutation and combination belongs to the statistics & probability cluster, where it shares terminology and assumptions with closely related tools.
Learning how permutation and combination is calculated
Combinations discount permutations by r! — the number of orderings within any single selected group. Working through the variables one at a time — n!, r — makes the result reproducible by hand and easier to sanity-check.
Using permutation and combination to make a decision
Lottery odds Poker probabilities Assigning committees or seats Because outputs depend on the assumptions you enter, run more than one scenario before committing to a figure.
How permutation and combination compares with related measures
distribution, dispersion, significance, sample all describe adjacent aspects of statistics & probability. Comparing this calculator's output against those measures — using the related tools listed on this page — prevents a single metric from being read in isolation.
Frequently asked questions
What's the difference in a real example?
Choosing 3 flavors from 10 for a scoop of ice cream is C(10,3) = 120. Choosing which flavor is bottom, middle, top is P(10,3) = 720.
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