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Permutation Calculator

· Last updated：2026-04-19

Calculate permutations with or without repetition, plus circular arrangements, with formulas and optional steps

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Written by the Elysia Tools team
Math502

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category
Math & Numbers
Input types
number, select, checkbox
Output type
json
Sample coverage
4
API ready
Yes

Overview

The Permutation Calculator is a fast and accurate math utility designed to compute the number of possible arrangements for a given set of items. Whether you are solving combinatorics problems, determining password combinations, or calculating seating arrangements, this tool supports permutations with repetition, without repetition, and circular arrangements, complete with step-by-step formulas.

When to use

  • When you need to find the exact number of ordered arrangements for a subset of items from a larger pool.
  • When solving probability or combinatorics assignments that require showing the underlying mathematical formulas and steps.
  • When calculating possible combinations for passwords, lock codes, or scheduling scenarios where order matters.

How it works

  1. 1Enter the total number of items available in your set.
  2. 2Specify the number of items you want to select and arrange.
  3. 3Choose the permutation type: without repetition, with repetition, or circular.
  4. 4Toggle the options to show the formula and calculation steps, then view the final computed result.

Use cases

  • Determining the number of possible outcomes for race podiums (1st, 2nd, and 3rd place).
  • Calculating the total number of possible PIN codes or passwords given a specific character set and length.
  • Figuring out seating arrangements for guests around a circular dining table.

Examples

1. Calculating Race Podium Outcomes

Math Student

Background

A student is working on a probability assignment involving a race with 8 runners.

Problem

Needs to find out how many different ways the gold, silver, and bronze medals can be awarded.

How to use

Set Total Items to 8, Selected Items to 3, and choose 'Without Repetition'. Enable 'Show Formula' to include in the homework.

Total Items: 8, Selected Items: 3, Permutation Type: without-repetition

Outcome

The tool outputs 336 possible arrangements and provides the formula P(8,3) = 8! / 5!.

2. Determining PIN Code Possibilities

Security Analyst

Background

An analyst is evaluating the brute-force vulnerability of a 4-digit PIN system.

Problem

Needs to calculate the total number of possible 4-digit PINs using numbers 0-9, where digits can be reused.

How to use

Set Total Items to 10, Selected Items to 4, and select 'With Repetition'.

FAQ

What is the difference between a permutation and a combination?

In a permutation, the order of the items matters (e.g., a lock code). In a combination, the order does not matter (e.g., selecting a team).

What does 'without repetition' mean?

It means once an item is selected from the pool, it cannot be chosen again. This is common in race rankings or seating arrangements.

How are circular permutations calculated?

Circular permutations account for arrangements in a circle where there is no fixed starting point. The formula used is typically (n-1)! for arranging n items.

Can I see the math behind the result?

Yes, by checking the 'Show Formula' and 'Show Steps' boxes, the calculator will output the exact mathematical formula and steps used to reach the answer.

Is there a limit to the numbers I can input?

The tool currently accepts inputs up to 100 for both the total items and selected items to prevent browser performance issues with massive factorials.

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Total Items: 10, Selected Items: 4, Permutation Type: with-repetition

Outcome

The calculator returns 10,000 possible permutations with the formula 10^4.

3. Planning a Circular Seating Arrangement

Event Planner

Background

An event planner is organizing a VIP dinner and needs to seat 6 guests around a round table.

Problem

Wants to know how many unique seating arrangements exist, keeping in mind that rotating the same circle of people doesn't count as a new arrangement.

How to use

Set Total Items to 6, Selected Items to 6, and choose 'Circular'.

Total Items: 6, Selected Items: 6, Permutation Type: circular

Outcome

The tool calculates 120 unique arrangements using the circular permutation formula (6-1)!.