Permutation
A permutation counts arrangements where changing the order creates a different result, such as first, second, and third place.
Free Combinatorics Tool
Permutation and Combination Calculator
Calculate nPr and nCr instantly. Enter the total number of objects and the number selected to compare arrangements where order matters and selections where order does not matter.
Enter Your Values
Use non-negative whole numbers where r is not greater than n.
Your Results
The calculated nPr and nCr values appear below.
Ready to Calculate
Enter n and r, choose a calculation type, and select the calculate button to see your results.
Core Concepts
Understand Permutations and Combinations
Both methods count possible outcomes, but the importance of order determines which calculation you should use.
Combination
A combination counts selections where order does not matter, such as choosing three members for a committee.
Factorial
A factorial multiplies a whole number by every positive whole number below it. For example, 5! equals 120.
Permutation Formula
Use a permutation when the position or arrangement of selected objects affects the final outcome.
Permutation formula
nPr = n! / (n − r)!
Permutation example
Suppose five runners compete for first, second, and third place. Since each finishing position is different, order matters.
5P3 = 5! ÷ 2! = 5 × 4 × 3 = 60
Therefore, the three positions can be filled in 60 different ways.
Combination Formula
Use a combination when you only need to select a group and the arrangement of its members does not matter.
Combination formula
nCr = n! / [r!(n − r)!]
Combination example
Suppose three people must be selected from a group of five. The order in which the people are selected does not create a new group.
5C3 = 5! ÷ [3! × 2!] = 10
Therefore, ten different groups of three people can be formed.
Simple Process
How to Use This Calculator
Follow these steps to calculate arrangements and selections without performing factorial calculations manually.
- Enter the value of n Add the total number of objects available in the complete set.
- Enter the value of r Add how many objects you want to arrange or select from the set.
- Select a calculation type Choose permutation, combination, or both calculations.
- Calculate the answer Select the calculate button to view the result and formula.
The value of r must not be greater
than n. Both inputs must be
non-negative whole numbers.
Quick Comparison
Permutation vs Combination
Ask whether rearranging the selected objects creates a new outcome. This helps you select the correct formula.
| Feature | Permutation | Combination |
|---|---|---|
| Does order matter? | Yes | No |
| Formula | n! ÷ (n − r)! | n! ÷ [r!(n − r)!] |
| Common example | Arranging winners by position | Selecting members for a team |
| Repeated arrangements | Different orders count separately | Different orders count as one group |
| Typical result | Usually larger | Usually smaller |
Common Questions
Frequently Asked Questions
Find clear answers about nPr, nCr, factorials, input requirements, and common real-world applications.
A permutation is an arrangement of objects where the order is important. ABC and BAC are considered two different permutations.
A combination is a selection of objects where the order is not important. ABC and BAC represent the same combination.
The value n represents the total number of available objects, while r represents how many objects are arranged or selected.
No. When selecting without replacement, r must be less than or equal to n because you cannot select more objects than are available.
Zero factorial, written as 0!, is equal to 1. This definition allows permutation and combination formulas to work consistently.
Use a permutation for rankings, seating arrangements, passwords, schedules, race positions, or any situation where changing the order creates a different outcome.
Use a combination for committees, teams, lottery selections, card hands, or other situations where only the selected group matters.