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Permutation & Combination Calculator

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n — Total Items r — Items Selected

Permutation & Combination Calculator

A Permutation & Combination Calculator is a free online tool that helps you calculate the number of possible arrangements and selections from a given set of items.

Permutations and combinations are closely related mathematical concepts, but they answer different questions.

A permutation counts arrangements where order matters.

A combination counts selections where order does not matter.

Calculator Pool’s free Permutation and Combination Calculator online lets you enter:

n = Total Items

and:

r = Items Selected

It then calculates both:

nPr (Permutation)

and:

nCr (Combination)

For example, when:

n = 10

and:

r = 3

the calculator gives:

10P3 = 720

and:

10C3 = 120

Important: Permutations and combinations assume the standard mathematical interpretation of selecting from a set of distinct items without replacement unless otherwise specified.

What Is a Permutation?

A permutation is an arrangement of objects where the order matters.

For example, suppose you have three letters:

A, B, C

If you choose two letters, the following are different permutations:

Because the order changes, AB and BA are counted separately.

This is the key idea behind permutations.

What Is a Combination?

A combination is a selection where the order does not matter.

Using the same letters:

A, B, C

If you choose two letters:

and:

represent the same combination.

Therefore, they are counted only once.

This difference between permutations and combinations is the most important concept to understand before using an nCr and nPr calculator. Our Permutation & Combination Calculator is free.

What Is a Permutation & Combination Calculator?

A permutation and combination calculator is an online math tool that calculates:

nPr

and:

nCr

using two inputs:

n — Total number of items

r — Number of items selected

Calculator Pool’s calculator displays both results and the formulas used to calculate them.

It is useful for:

Mathematics
Probability
Statistics
Combinatorics
School assignments
Competitive exams
Counting problems

Kindly refer our Free Permutation & Combination Calculator online with your friends and family.

How to Use the Permutation & Combination Calculator

Using Calculator Pool’s permutation calculator online is simple.

Step 1: Enter n

Enter the total number of available items.

For example:

Step 2: Enter r

Enter the number of items you want to select or arrange.

For example:

Step 3: Calculate

Click:

Calculate nPr & nCr

The calculator will display:

Permutation (nPr) = 720

Combination (nCr) = 120

It also shows the mathematical formulas and explains the difference between the two.

Permutation Formula

The standard permutation formula is:

nPr = n! / (n − r)!

Where:

n = total number of items
r = number of items selected
! = factorial

A permutation counts the different possible arrangements of r items selected from n items. Permutation & Combination Calculator is mobile friendly.

Combination Formula

The standard combination formula is:

nCr = n! / [r!(n − r)!]

Where:

n = total number of items
r = number selected
! = factorial

A combination counts selections where the order of the selected items does not matter. Our Permutation & Combination Calculator is free for all our users.

What Is Factorial?

A factorial is the product of all positive integers from a number down to 1.

For example:

5! = 5 × 4 × 3 × 2 × 1

Therefore:

5! = 120

Another example:

3! = 3 × 2 × 1 = 6

Factorials are fundamental to both permutation and combination formulas.

Permutation Example

Suppose you have:

5 students

and want to select and arrange:

3 students

The number of permutations is:

5P3 = 5! / (5 − 3)!

= 5! / 2!

= 120 / 2

= 60

Therefore:

5P3 = 60

Combination Example

Using the same values:

n = 5

r = 3

The number of combinations is:

5C3 = 5! / [3!(5 − 3)!]

= 120 / (6 × 2)

= 10

Therefore:

5C3 = 10

Notice:

5P3 = 60

but:

5C3 = 10

The difference occurs because permutations distinguish between different orders, while combinations do not. Use our Permutation & Combination Calculator for free.

nPr vs nCr

The easiest way to remember the difference is:

nPr

Order matters

nCr

Order does not matter

For example, selecting:

A, B

and:

B, A

creates two different permutations.

But it represents only one combination.

Permutation vs Combination Example

Suppose you have:

A, B, C

and want to choose two.

Permutations

The possible arrangements are:

Therefore:

3P2 = 6

Combinations

The possible selections are:

Therefore:

3C2 = 3

This example clearly demonstrates why permutations are larger than combinations when:

0 < r < n

Relationship Between nPr and nCr

A useful relationship is:

nPr = nCr × r!

For example:

10P3 = 10C3 × 3!

= 120 × 6

= 720

Therefore:

10P3 = 720

This relationship can be used to check calculations.

Permutation and Combination Table

n	r	nPr	nCr
5	2	20	10
5	3	60	10
6	2	30	15
6	3	120	20
8	2	56	28
10	3	720	120
10	4	5,040	210

When Should You Use a Permutation?

Use a permutation when the order or position matters.

Examples include:

Arranging people in seats
Ranking competitors
Creating ordered codes
Assigning positions
Arranging books
Creating ordered sequences

For example, first place and second place are different positions, so ranking competitors is a permutation-type problem.

When Should You Use a Combination?

Use a combination when you are simply choosing items and order does not matter.

Examples include:

Selecting a committee
Choosing team members
Picking lottery numbers
Selecting books
Choosing menu items
Selecting a group from a larger set

For example, a committee containing Alice, Bob, and Carol is the same committee regardless of the order in which their names are listed.

Permutation Calculator for Arrangements

A permutation calculator is useful when a problem asks:

“How many different ways can these items be arranged?”

Suppose 4 books are being arranged on a shelf.

The number of arrangements is:

4! = 24

because all 4 positions matter.

Combination Calculator for Selections

A combination calculator is useful when a problem asks:

“How many different groups can be selected?”

Suppose you have 10 people and want to select 3 for a committee.

The calculation is:

10C3 = 120

The order of the people chosen does not matter.

nCr Calculator

An nCr calculator calculates the number of combinations.

Formula:

nCr = n! / [r!(n − r)!]

For example:

10C3 = 120

This means there are 120 different ways to select 3 items from 10 when order is irrelevant.

nPr Calculator

An nPr calculator calculates permutations.

Formula:

nPr = n! / (n − r)!

For:

10P3

the result is:

720

This means there are 720 different ordered arrangements of 3 items selected from 10.

Permutation and Combination in Probability

Permutations and combinations are frequently used in probability calculations.

For example, combinations can help determine the number of possible groups or selections, while permutations can count ordered outcomes.

Suppose you are selecting 3 people from 10:

10C3 = 120

If the order of those 3 people matters:

10P3 = 720

These counts can then form part of a probability calculation.

Calculator Pool also provides a dedicated Probability Calculator for basic probability problems.

Permutation and Combination in Statistics

Combinations and permutations are important concepts in statistics and probability.

They can be used when determining:

Possible samples
Group selections
Ordered samples
Experimental outcomes
Possible arrangements

Understanding whether order matters is essential before selecting an nPr or nCr calculation.

Permutation and Combination in Mathematics

Permutation and combination belong to the area of mathematics known as combinatorics.

Combinatorics studies ways of:

Counting objects
Arranging objects
Selecting groups
Constructing sequences
Analyzing possible outcomes

The formulas for nPr and nCr provide efficient ways to count large numbers of possibilities.

Permutations With All Items

When:

r = n

the permutation formula becomes:

nPn = n!

For example:

5P5 = 5!

= 120

This represents arranging all five items.

Combinations With All Items

When:

r = n

the combination becomes:

nCn = 1

There is only one way to select all available items as a group.

For example:

5C5 = 1

Choosing No Items

When:

r = 0

the standard formulas give:

nP0 = 1

and:

nC0 = 1

There is one way to select or arrange zero items: select nothing.

Permutations When r = 1

If:

r = 1

then:

nP1 = n

because there are n possible choices for the single ordered position.

Similarly:

nC1 = n

Symmetry of Combinations

Combinations have the property:

nCr = nC(n − r)

For example:

10C3 = 10C7

Both equal:

120

This happens because selecting 3 items to include is equivalent to selecting 7 items to exclude.

Why Is nCr Smaller Than nPr?

When:

0 < r < n

every combination corresponds to:

different permutations.

Therefore:

nPr = nCr × r!

Since:

r! ≥ 1

the permutation count is generally at least as large as the combination count.

Permutations With Repetition

The standard nPr formula assumes that you are selecting distinct items without replacement.

If repetition is allowed, a different formula may be required.

For example, a 3-digit code using digits 0–9 with repetition allowed has:

10 × 10 × 10 = 1,000

possible codes.

This is not the same calculation as ordinary nPr without repetition.

Combinations With Repetition

Combinations where repetition is allowed also require a different formula from the standard nCr calculation.

The common formula for selecting r items from n types with repetition allowed is:

C(n + r − 1, r)

This is also known as the stars and bars approach in combinatorics.

Calculator Pool’s current permutation and combination calculator handles the standard nPr and nCr formulas without repetition.

Why Does Order Matter?

Order matters whenever positions, sequence, ranking, or arrangement changes the meaning of the result.

For example:

Gold, Silver, Bronze

is different from:

Silver, Gold, Bronze

because the positions are different.

This makes ranking a permutation problem.

Why Doesn’t Order Matter in Combinations?

A combination represents a group rather than an arrangement.

Suppose:

Alice + Bob + Carol

are selected for a team.

Writing the team as:

Carol + Alice + Bob

does not create a new team.

Therefore, order is ignored in combinations.

Permutation and Combination Word Problems

A useful way to identify the correct calculation is to look at the wording.

Words suggesting permutation include:

Arrange
Order
Rank
Position
Sequence
Assign

Words suggesting combination include:

Select
Choose
Group
Committee
Team
Set

The key question is:

Does changing the order create a different result?

If yes, think permutation.

If no, think combination.

Permutation Example: Seating Arrangement

Suppose 5 people need to sit in 5 different seats.

The number of arrangements is:

= 120

Every person’s position matters, so this is a permutation problem.

Combination Example: Selecting a Team

Suppose 10 students are available and 4 must be selected for a team.

The number of possible teams is:

10C4

= 210

The order in which the team members are selected does not matter.

Permutation Example: Ranking

Suppose 8 competitors participate in a contest and you want to know how many possible arrangements there are for first, second, and third place.

Here:

n = 8

r = 3

Therefore:

8P3 = 8 × 7 × 6

= 336

The order matters because first, second, and third are different positions.

Combination Example: Committee Selection

Suppose there are:

8 people

and:

3 committee members

need to be selected.

Because there are no positions within the committee:

8C3 = 56

Thus, there are:

56 possible committees

Permutation and Combination Examples

Example 1

n = 6, r = 2

Permutation:

6P2 = 30

Combination:

6C2 = 15

Example 2

n = 7, r = 3

Permutation:

7P3 = 210

Combination:

7C3 = 35

Example 3

n = 10, r = 4

Permutation:

10P4 = 5,040

Combination:

10C4 = 210

Permutation & Combination Calculator for Students

A permutation and combination calculator for students can be useful for checking homework and practicing combinatorics.

It can help students understand:

Factorials
nPr
nCr
Order matters
Order does not matter
Counting principles
Probability

It can also help verify manual calculations.

Permutation & Combination Calculator for Probability

Students studying probability often encounter nPr and nCr.

For example, a probability problem may require calculating:

Number of favorable combinations

divided by:

Total possible combinations

The correct selection of nPr or nCr depends on whether order matters.

Permutation & Combination Calculator for Competitive Exams

Permutation and combination problems are common in many mathematics curricula and competitive-exam preparation.

An nCr and nPr calculator can help students verify answers after solving problems manually.

However, understanding the underlying formulas remains important because calculators cannot determine the intended interpretation of a word problem for you.

Common Permutation and Combination Mistakes

Confusing nPr and nCr

Ask whether order matters.

Using the Wrong r Value

Make sure r represents the number of items selected.

Entering r Greater Than n

Standard nPr and nCr require:

0 ≤ r ≤ n

Forgetting Factorials

Both formulas involve factorials.

Treating a Group as an Arrangement

A committee selection is generally a combination unless positions within the group matter.

Assuming Repetition Is Allowed

Standard nPr and nCr calculations assume selection without replacement.

What Happens When r Is Greater Than n?

For standard nPr and nCr:

r cannot be greater than n

For example:

5C7

is not defined under the standard selection interpretation because you cannot choose 7 distinct items from a set of only 5 items.

Calculator Pool validates the input and requires:

0 ≤ r ≤ n

Large Permutation and Combination Numbers

Permutation and combination results can become extremely large.

For example:

20!

is already a very large number.

As n increases, factorial-based values grow extremely quickly.

Calculator Pool therefore limits n to a safe range for exact JavaScript calculations.

Why Use an Online Permutation Calculator?

An online permutation calculator can save time when values are large.

It can:

Calculate nPr instantly
Calculate nCr instantly
Avoid manual factorial calculations
Show formulas
Compare permutations and combinations
Reduce arithmetic mistakes

Benefits of Calculator Pool’s Permutation & Combination Calculator

Calculator Pool’s free permutation and combination calculator online lets you calculate both results from the same two inputs.

Enter:

and:

The calculator provides:

nPr

nCr

It also explains:

Permutation Formula

Combination Formula

and:

nPr = nCr × r!

This makes it useful for both calculation and learning.

Factorial Calculator Relationship

A factorial is fundamental to permutation and combination calculations.

For example:

6! = 720

Then:

6P3 = 6! / 3!

= 720 / 6

= 120

Understanding factorials makes nPr and nCr formulas easier to follow.

Permutation and Combination vs Probability

These concepts are related but not identical.

Permutation

Counts ordered arrangements.

Combination

Counts unordered selections.

Probability

Measures how likely an event is.

A probability problem may use permutations or combinations to count possible outcomes before calculating probability.

Frequently Asked Questions

What is a Permutation & Combination Calculator?

A Permutation & Combination Calculator calculates the number of possible permutations (nPr) and combinations (nCr) from n total items and r selected items.

What is permutation?

Permutation counts arrangements where order matters.

What is combination?

Combination counts selections where order does not matter.

What is nPr?

nPr is the number of ways to arrange r items from n items.

nPr = n! / (n − r)!

What is nCr?

nCr is the number of ways to choose r items from n items when order does not matter.

nCr = n! / [r!(n − r)!]

What is the difference between nPr and nCr?

nPr considers order, while nCr ignores order.

What does n mean?

n represents the total number of available items.

What does r mean?

r represents the number of items selected or arranged.

What is factorial?

Factorial is the product of all positive integers from a number down to 1.

For example:

5! = 120

What is 10P3?

10P3 = 720

What is 10C3?

10C3 = 120

Why is permutation greater than combination?

For the standard case where 0 < r < n, every combination corresponds to r! different permutations.

When should I use nPr?

Use nPr when changing the order produces a different outcome.

When should I use nCr?

Use nCr when the selected group is the same regardless of order.

Can r be greater than n?

No. Standard nPr and nCr require:

0 ≤ r ≤ n

What happens when r = 0?

Both:

nP0 = 1

and:

nC0 = 1

What happens when r = n?

nPn = n!

and:

nCn = 1

What is the relationship between nPr and nCr?

nPr = nCr × r!

Can permutations be used for seating arrangements?

Yes. Seating arrangements are usually permutation problems because position matters.

Can combinations be used for team selection?

Yes. Selecting a team generally uses combinations when member order does not matter.

Are nPr and nCr used in probability?

Yes. They can be used to count possible outcomes in probability problems.

Are repetitions allowed in standard nPr and nCr?

No. Standard formulas assume distinct items selected without replacement.

Can permutations have repetition?

Yes, but repeated-selection problems require different formulas depending on the setup.

Can combinations have repetition?

Yes, but a different combination formula is required.

What is the difference between permutations with repetition and without repetition?

Without repetition, each selected item can be used only once. With repetition, items may be selected more than once, requiring a different counting formula.

What is a combination with repetition?

It is a selection where the same type of item can be chosen multiple times. A common formula is:

C(n + r − 1, r)

Is nCr the same as nC(n − r)?

Yes.

nCr = nC(n − r)

Why does nCr equal nC(n − r)?

Choosing r items to include is equivalent to choosing n − r items to leave out.

Can I use decimal values for n and r?

No. Standard nPr and nCr require whole-number inputs.

Can I use negative values?

No. The standard calculator requires:

n ≥ 0

and:

0 ≤ r ≤ n

Why do permutation and combination values become so large?

Factorials grow extremely quickly, so nPr and nCr can become very large even for moderately sized n.

Is the Permutation & Combination Calculator free?

Yes. Calculator Pool’s Permutation & Combination Calculator is free to use online.

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Factorial Calculator

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Standard Deviation Calculator

Analyze the spread and variability of numerical datasets.

Permutation & Combination Calculator – Quick Summary

A Permutation & Combination Calculator helps you calculate the number of possible arrangements and selections from a set of items.

The two primary calculations are:

Permutation (nPr)

where:

Order Matters

and:

Combination (nCr)

where:

Order Does Not Matter

The permutation formula is:

nPr = n! / (n − r)!

The combination formula is:

nCr = n! / [r!(n − r)!]

A useful relationship between them is:

nPr = nCr × r!

For example, with:

n = 10

r = 3

the results are:

10P3 = 720

10C3 = 120

Use Calculator Pool’s free permutation and combination calculator online to calculate nPr and nCr quickly, compare the results, and understand whether a mathematical problem requires an ordered arrangement or an unordered selection.

These concepts are widely used in combinatorics, probability, statistics, mathematics, and competitive-exam problems.