Permutation Calculator

What Is a Permutation Calculator?

A permutation calculator determines the number of possible ordered arrangements that can be made from a given number of items. Unlike combinations, permutations take the order of the selected items into account. For example, arranging A, B and C as ABC, ACB and BAC produces different permutations because the order changes.

This free online permutation calculator can calculate standard permutations without repetition as well as permutations where repetition is allowed. It uses whole-number values for n and r and produces an exact integer result, making it useful for mathematics, probability, statistics, computer science and other situations involving ordered arrangements.

Why Use Permutation Calculator?

Calculating permutations manually can become increasingly difficult as the numbers become larger. This calculator performs the calculation instantly while showing the formula used, helping you obtain and check permutation results quickly.

How to Use Permutation Calculator

Use the calculator whenever you need to determine how many ordered arrangements are possible from a set of items.

1. Enter n — Enter the total number of available items.
2. Enter r — Enter how many items will be selected and arranged.
3. Choose repetition — Select whether items can be used more than once.
4. Calculate the result — Select Calculate Permutations to display the number of possible ordered arrangements and the formula used.

How to Calculate Permutations

A permutation is an arrangement in which order matters. For permutations without repetition, the standard formula is nPr = n! / (n - r)!. Here, n represents the total number of available items and r represents the number of items being arranged. The formula can also be written as a multiplication of consecutive descending values: n × (n − 1) × (n − 2) and so on for r factors.

For example, if there are 10 available items and 3 are arranged at a time, the number of permutations is 10 × 9 × 8 = 720. This means there are 720 different ordered arrangements of three items selected from the ten available items.

Permutations Without Repetition

When repetition is not allowed, an item can only appear once within each arrangement. The number of available choices therefore decreases after each selection. For n = 5 and r = 3, the calculation is 5 × 4 × 3 = 60. The same result can be obtained from 5! / (5 − 3)!.

Permutations With Repetition

When repetition is allowed, each position can be filled from all n available choices. The number of arrangements is therefore n^r. For example, if there are 5 possible symbols and 3 positions, repetition allowed produces 5³ = 125 possible arrangements.

Permutation Calculator Results and Limitations

The calculator accepts non-negative whole numbers for n and r. For permutations without repetition, r cannot be greater than n because there cannot be more unique selections than available items. A value of zero is also supported: there is one way to arrange zero items, represented mathematically by 0! = 1.

The calculator performs its calculations locally in your browser using JavaScript and does not require a server-side calculation. It uses BigInt arithmetic for the permutation calculation, allowing very large whole-number results to be calculated without the rounding associated with ordinary JavaScript floating-point numbers. BigInt calculations are intended for integers only, so decimal or fractional values are not valid inputs for this calculator.

Frequently Asked Questions