Probability

Permutation Calculator

Calculate nPr for ordered selections, compare it with the corresponding unordered combination count, show the number of orderings inside each combination, and calculate an expected success count from separately entered independent trials and success probability.

Available items (n)Ordered selections (r)Number of independent trialsSuccess probability per ordering (%)

Number of permutations-

Unordered combinations-

Order arrangements per combination-

Expected successful trials-

Decision view

Unordered choices multiplied by orderings

Unordered choices multiplied by orderingsThe counting identity C(n,r) × r! = P(n,r) separates which items are selected from how each selected set can be arranged.

Exact scenario comparisonOrdered selections (r) changes while all other entered assumptions remain constant.

Ordered selections (r)	Number of permutations	Unordered combinations	Order arrangements per combination	Expected successful trials

How to use Permutation Calculator

Enter the number of distinct available items n and the number selected r, ensuring that both are whole numbers with zero less than or equal to r less than or equal to n.
Use permutations when sequence or assigned position changes the outcome; use the displayed combination count when only membership matters.
Treat the expected successful-trial result as a separate probability calculation and verify that the entered trial probability and independence assumption are appropriate.

Calculator guide

Understanding Permutation Calculator

Permutations count ordered selections: choosing A then B is different from choosing B then A. The calculation therefore depends on distinct items, selection without replacement, and an ordered outcome definition.

Order changes identity Permutations count the same selected items multiple times when their sequence or assigned roles differ.

Without replacement Once an item is selected, it is not available for another position in the same ordering.

Combination bridge nPr equals nCr multiplied by r factorial.

Rapid growth Ordered counts can become extremely large after only modest increases in n or r.

Calculation method

How the calculation works

Calculate n! divided by (n-r)! for ordered selections and compare it with combinations and the ordering factor. Calculate n! divided by (n-r)! for ordered selections and compare it with combinations and the ordering factor.

Counting rule

Choose the structure before applying a formula

Most counting errors come from using the right arithmetic for the wrong outcome definition.

Order matters, no repeats Use permutations nPr, as modeled on this page.

Order does not matter Use combinations nCr so each selected group is counted once.

Order matters, repeats allowed Use a with-replacement rule such as n raised to r when every position can reuse any item.

Identical or constrained items Use a problem-specific adjustment for duplicates, required positions, exclusions, adjacency, or circular equivalence.

Define what makes two outcomes different before counting them.

Worked situations

Practical examples

Selecting four ordered positions from 12 distinct items gives 12 times 11 times 10 times 9, or 11,880 permutations.
The same selection contains 495 unordered combinations, and each four-item combination has 4! = 24 possible internal orders; 495 times 24 equals 11,880.
Ten independent trials with a 5% success probability have an expected success count of 0.5, which is a long-run average rather than a promise of half a success.

Better inputs

Useful tips

Write one swapped example—such as AB versus BA—to decide whether order genuinely creates a different outcome.
If items can repeat, use a with-replacement counting rule rather than nPr.
Keep counting and probability assumptions separate; the number of arrangements does not determine an arrangement's success probability unless the random mechanism is specified.

Before relying on the result

Limitations and common mistakes

The formula assumes distinct items selected without replacement and treats every ordering as a separate count.
It does not handle repeated identical objects, circular arrangements, restricted positions, block constraints, or combinations with replacement.
Expected successes use a constant entered probability across independent trials and do not describe the full binomial distribution or outcome uncertainty.

Reference

Key terms

Permutation: An ordered selection of r distinct items from n available items.
Combination: An unordered selection in which membership matters but arrangement does not.
Factorial: Product of positive integers from a given whole number down to one, with 0! defined as one.
Expected value: Long-run average outcome across repetitions of the stated probability model.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

What if r is greater than n?

An ordered selection without replacement is impossible. Enter r no greater than n.

Why is the permutation count larger than the combination count?

Each unordered r-item combination is expanded into r factorial different internal orders.

Are all permutations equally likely?

Only if the generating process gives every ordering equal probability. The counting formula alone does not establish likelihood.

Can expected successes be fractional?

Yes. Expected value is a long-run average across repeated sets of trials, not a literal result required in one set.