Quick answer
Permutations count ordered selections, nPr = n! ÷ (n − r)!, and combinations count unordered ones, nCr = n! ÷ (r!(n − r)!). Choosing 3 of 10 gives 720 permutations and 120 combinations.
Enter the total items n and the chosen items r. The calculator returns nCr, nPr and n!.
Related tools: Probability Calculator: Probability of A and B, and A or B, for independent or mutually exclusive events. Dice Average Calculator: Average, minimum and maximum of rolling several dice. Expected Value Calculator: Expected value and variance of a discrete distribution.
How to use the Permutation and Combination Calculator
- Enter total items (n) (a whole number, between 0 and 1000).
- Enter chosen items (r) (a whole number, between 0 and 1000).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Total items (n) | A whole number | ≥ 0, ≤ 1000 |
| Chosen items (r) | A whole number | ≥ 0, ≤ 1000 |
Formula and method
Formula
nPr = n! ÷ (n − r)!; nCr = n! ÷ (r!(n − r)!)
IEEE-754 double precision; shown to at most 6 decimal places.
Exact BigInt arithmetic.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Total items (n) = 10; Chosen items (r) = 3.
| Result | Value |
|---|---|
| Combinations (nCr) | 120 |
| Permutations (nPr) | 720 |
| n! | 3,628,800 |
Example 2
Inputs: Total items (n) = 52; Chosen items (r) = 5.
| Result | Value |
|---|---|
| Combinations (nCr) | 2,598,960 |
| Permutations (nPr) | 311,875,200 |
| n! | 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 |
Example 3
Inputs: Total items (n) = 5; Chosen items (r) = 0.
| Result | Value |
|---|---|
| Combinations (nCr) | 1 |
| Permutations (nPr) | 1 |
| n! | 120 |
The first line is the headline result (combinations (nCr)). Below it the permutation and Combination Calculator lists permutations (nPr) and n!, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Order or not
A podium of 1st, 2nd and 3rd from 10 runners is a permutation (10 × 9 × 8 = 720). A committee of 3 from 10 people is a combination: 720 ÷ 3! = 120.
Where this estimate stops applying
r can't exceed n, and 0! = 1. Combinations have the symmetry nCr = nC(n − r). 10! = 3,628,800.
How the result changes with total items
Holding every other input at the first worked example, this table changes only total items. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Total items | Combinations (nCr) | Permutations (nPr) | n! |
|---|---|---|---|
| 5 | 10 | 60 | 120 |
| 10 | 120 | 720 | 3,628,800 |
| 20 | 1,140 | 6,840 | 2,432,902,008,176,640,000 |
| 40 | 9,880 | 59,280 | 815,915,283,247,897,734,345,611,269,596,115,894,272,000,000,000 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Chosen items (r): entering "4" shows "r cannot be greater than n." beside that field and no result.
Choosing between the Permutation and Combination Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Probability Calculator](/statistics/probability-calculator/) | p, p and events are | P(A and B) |
| [Dice Average Calculator](/statistics/dice-average/) | number of dice and sides per die | Average roll |
| [Expected Value Calculator](/statistics/expected-value/) | values and probabilities | Expected value E[X] |
| [Bayes' Theorem Calculator](/statistics/bayes-theorem-calculator/) | prior P, p and p | P(A | B) |
| [Implied Probability Calculator](/statistics/implied-probability/) | odds format and odds | Implied probability |
Background: probability and distributions
A probability model only applies when its conditions hold. The binomial model needs a fixed number of independent trials with two outcomes and a constant success probability; its mass function gives the chance of exactly k successes. The normal density is f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²)).
A cumulative probability adds up everything at or below a value, a density or mass function gives the value at a point. For a standard normal variable, z = −1.96 leaves 0.025 in the lower tail, which is why ±1.96 bounds a 95% interval.
Assumptions and limitations
- r cannot exceed n
- 0! = 1
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for chosen items (r) to confirm this.
Frequently asked questions
What is 5C2?
5C2 = 5! ÷ (2! × 3!) = 10.
When do I use permutations?
Use permutations when order matters, such as rankings on a podium; use combinations when it doesn't, such as a committee.
What is 0!?
0! = 1 by definition, which keeps formulas such as nC0 = 1 consistent.
What does the Permutation and Combination Calculator calculate?
Number of permutations and combinations of r items from n. It returns Combinations (nCr), Permutations (nPr) and n!.
What formula does the Permutation and Combination Calculator use?
nPr = n! ÷ (n − r)!; nCr = n! ÷ (r!(n − r)!). Exact BigInt arithmetic.
What do I need to enter in the permutation and Combination Calculator?
You enter Total items (n) and Chosen items (r).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. r cannot exceed n.
What are the limits of this calculator?
r cannot exceed n. 0! = 1.
What is an example of a permutation and Combination result?
With total items (n) 10 and chosen items (r) 3, the combinations (nCr) is 120.
Related calculators
- Probability Calculator — Probability of A and B, and A or B, for independent or mutually exclusive events.
- Dice Average Calculator — Average, minimum and maximum of rolling several dice.
- Expected Value Calculator — Expected value and variance of a discrete distribution.
- Bayes' Theorem Calculator — Posterior probability from a prior and test accuracy.
- Implied Probability Calculator — Probability implied by betting odds.
- Standard Deviation Calculator — Sample or population standard deviation with variance and mean.
Sources
- Standard statistics definition (e.g. NIST/SEMATECH e-Handbook of Statistical Methods)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.