Mathematics calculator

Permutation Calculator

Compare ordered selections without repetition, P(n,r), with ordered sequences where repetition is allowed, nʳ.

Calculator inputs

Enter your values

Ordered arrangements
Calculated output

Result summary

Your result will appear here

Complete the required fields and select the calculate button.

Table of Contents

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What is Permutation?

The Permutation Calculator counts ordered selections from a set of distinct objects. It provides two answers from the same values: permutations without repetition and ordered selections with repetition allowed. The difference matters whenever choosing an item removes it from future positions.

Order is central to both results. Choosing A then B is different from choosing B then A. Examples include assigning first, second and third place, arranging letters in available positions or creating a sequence of symbols. If order does not matter, a combination count answers a different question.

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How to Use This Calculator

Enter n, the total number of distinct choices, and r, the number of ordered positions to fill. Both inputs must be whole numbers from 0 through 100, and r cannot exceed n. Choose Calculate to receive the count without repetition and the count when repetition is allowed.

Before using the results, translate the real situation into the page’s assumptions. For the first result, each of the n objects is distinct and can be selected at most once. For the repetition result, every one of the n choices is available again for every position. Neither result adjusts for identical objects, special exclusions or rules that change the choices from one position to another.

The answers can grow quickly. The page limits n to 100 so its ordinary numeric arithmetic remains finite. Large displayed values may use scientific notation supplied by the browser even though the mathematical count is a whole number.

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Formula Used

Without repetition: P(n,r) = n! ÷ (n − r)!

With repetition: nr

The factorial symbol means multiplying the positive whole numbers down to 1, and 0! is defined as 1. These formulas give 1 when r = 0 because there is exactly one empty ordered selection.

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Variables Explained

SymbolMeaning on this pageAllowed values
nTotal number of distinct objects or choicesWhole number from 0 to 100
rNumber of ordered positions being filledWhole number from 0 through n
P(n,r)Ordered selections without reusing an objectWhole-number count
nrOrdered sequences when every choice may be reusedWhole-number count

“Distinct” means the objects can be told apart. Five differently labeled cards fit the formula. Five cards containing repeated labels may require dividing for indistinguishable arrangements or using a separate multiset method.

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Worked Example

Take five finalists and fill two ranked positions. Without repetition, there are five choices for first place and four remaining choices for second place:

P(5,2) = 5! ÷ 3! = 5 × 4 = 20

If the same choice may appear in both positions, there are five choices each time:

52 = 25

The calculator returns 20 permutations and 25 permutations with repetition. The extra five repeated sequences are AA, BB, CC, DD and EE. That difference is a quick check that the two formulas are answering separate versions of the question.

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Why Use This Calculator?

The result is a count of possible ordered outcomes, not a probability. To calculate probability, you also need the number of outcomes belonging to the event and a reason to treat the outcomes as equally likely. A count of 20 possible rankings does not mean each ranking occurs with probability 20%.

Check whether positions truly create different outcomes. A president and treasurer assignment is ordered because swapping the people changes their roles. Choosing two people for an otherwise identical committee is unordered, so the permutation result would count each pair twice. Also decide whether reuse is possible: digits in a PIN may repeat, while winners in a race cannot occupy two finishing places.

Counts increase rapidly as either input rises. A large answer is normal and does not by itself indicate a calculation error. Reproduce a small case by listing its outcomes when you are unsure which model fits.

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Common Mistakes

  • Using permutations for a group where order has no meaning.
  • Reading “with repetition” as repeated identical objects; here it means a choice may be used again in another position.
  • Entering r larger than n for the no-repetition case.
  • Entering decimal values even though counts of objects and positions must be whole numbers.
  • Assuming the formula handles restrictions such as “A cannot be first.”
  • Treating the number of arrangements as an event probability.
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Frequently Asked Questions

When does order matter?

Order matters when swapping selected objects produces a different outcome. Race positions, job roles and character sequences are ordered. An ordinary group with no roles is generally unordered.

What does repetition mean here?

It means a selected choice becomes available again for the next position. A four-digit code can reuse a digit, while a four-person finishing order cannot reuse a competitor.

Why must r be no greater than n?

The page always calculates the no-repetition result, which cannot fill more positions than there are distinct available objects. Even if only the repetition result interests you, both outputs share the validated inputs.

What happens when r is zero?

Both formulas return one. In counting language, there is one way to choose the empty ordered sequence. This convention also keeps factorial identities consistent.

Can this count arrangements of repeated letters?

Not generally. The no-repetition formula assumes the starting objects are distinguishable. A word with identical letters needs a multiset-permutation adjustment so swapping identical copies is not counted as a new arrangement.

Why is the answer sometimes displayed in scientific notation?

Permutation counts become enormous quickly. A browser may shorten a large numeric response using powers of ten. That display is an approximation format for the magnitude, while the underlying combinatorial count is an integer.

Is a combination smaller than the matching permutation?

For the usual distinct-object case, each unordered selection corresponds to r! orders, so P(n,r) equals C(n,r) × r!. They are equal when r is zero or one and otherwise the permutation count is larger.

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Sources & References