math calculator

Permutation & Combination Calculator

Calculate permutations and combinations with or without repetition, and generate all possible subsets.

Save this combinatorics workspace

Keep this calculator ready for estimating game probabilities, calculating password configurations, or planning sample sizes.

Saved runs

Inputs

Total Items (n)

Chosen Items (r)

Allow repetition of items

Active Output Display

Results

Combinations Result

Permutations (P)60

Combinations (C)10

Factorials Reference

n! (5!) = 120

r! (3!) = 6

(n - r)! (2!) = 2

Subsets List (10 total)

ABCABDABEACDACEADEBCDBCEBDECDE

Using elements: A, B, C, D, E

Formula

Permutations & Combinations Math

Permutations (no rep): P(n,r) = n! / (n-r)!
Combinations (no rep): C(n,r) = n! / (r!(n-r)!)

Permutations count ordered arrangements where order is critical. Combinations count unordered groups where order is irrelevant.

Step by step

How the calculation works

1Input total items (n) and selections size (r).
2Toggle repetition rules checkbox to allow/disallow item reuse.
3Select calculation mode to choose the highlighted main output metric.
4The factorials card shows calculations for n!, r!, and (n-r)!.
5If n and r are 5 or less, the subsets panel generates and displays all exact subsets alphabetically.

Example

Combinatorics Example scenario

Choosing 3 names from 5 candidates: Combinations = 5! / (3! × 2!) = 10 ways. Permutations (arranging them in 1st, 2nd, 3rd place) = 5! / 2! = 60 ways.

Related guides

Learn more about this calculation

Guide

Standard Deviation: Measuring Data Spread and Variability

Understand standard deviation, how it measures data spread, the difference between population and sample standard deviation, and how to interpret results in real-world contexts.

Guide

Random Number Generation: True Random vs Pseudo-Random Numbers

Explore random number generation methods, true randomness vs pseudo-randomness, common algorithms, and applications in simulations, games, and cryptography.

Permutations vs Combinations Growth

Observe how permutations (ordered choices) scale much faster than combinations (unordered choices) as selection size increases.

How to understand and use this calculator

Foundations of Combinatorics: The Science of Selection and Arrangement

Combinatorics is a branch of mathematics concerned with counting, arranging, and selecting objects from a finite set. It answers fundamental questions such as: "How many different lottery combinations are possible?", "How many secure passwords can be generated?", or "In how many ways can a committee of three be selected from a group of ten people?"

To solve these problems, combinatorics relies on two basic operations: permutations and combinations. The core distinction between them lies in one simple rule: order.

- In Permutations, the order of arrangement is critical. For example, the sequence [A, B, C] is distinct from [C, B, A]. Common real-world examples include safe combination dials, racetrack finishes, and telephone numbers.

- In Combinations, the order does not matter. The subset containing {A, B, C} is identical to the subset containing {C, B, A}. Examples include poker hands, lottery tickets, and picking a team from a pool of candidates.

The Fundamental Counting Principle and Factorials

Before exploring formal combinatorial formulas, we must understand the Fundamental Counting Principle. If one task can be completed in \(p\) ways, and a subsequent task can be completed in \(q\) ways, then both tasks can be performed in \(p \times q\) ways.

This leads directly to the concept of Factorials. The factorial of a positive integer \(n\) (written as \(n!\)) is the product of all positive integers less than or equal to \(n\):

\[n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1\]

By convention, the factorial of zero is defined as exactly 1 (\(0! = 1\)). Factorials scale extremely fast: \(5! = 120\), \(10! = 3,628,800\), and \(70!\) exceeds the total number of atoms in the observable universe. When calculating permutations and combinations, we simplify these massive numbers by cancelling identical terms in the numerator and denominator fractions.

Permutation Formulas: With and Without Repetition

We categorize permutations based on whether we are allowed to select the same item more than once (repetition):

1. Permutations Without Repetition:

You have \(n\) unique items and select \(r\) of them without replacement. The number of unique arrangements is given by:

\[P(n, r) = \frac{n!}{(n-r)!}\]

For instance, if you have 5 books and want to arrange 3 of them on a shelf, \(P(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60\) ways.

2. Permutations With Repetition:

Each time you make a choice from \(n\) items, you return the selected item to the pool. The number of arrangements is simply:

\[P_R(n, r) = n^r\]

This applies to lock codes or passwords. If a lock has a 3-digit dial and each dial has 10 digits (0-9), the number of permutations is \(10^3 = 1000\).

Combination Formulas: With and Without Repetition

Combinations are similarly classified by whether items can be repeated:

1. Combinations Without Repetition (Binomial Coefficient):

Selecting \(r\) items from \(n\) items without replacement, where order is irrelevant. The formula is:

\[C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}\]

For example, if you choose a committee of 3 people from a pool of 5 candidates, \(C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{120}{6 \times 2} = 10\) ways. This is also written as the binomial coefficient "n choose r".

2. Combinations With Repetition (Stars and Bars Theorem):

Selecting \(r\) items from \(n\) categories where each category has an infinite supply (e.g., choosing 5 sodas from a choice of 3 brands). The number of combinations is computed using the Stars and Bars theorem:

\[C_R(n, r) = \binom{n+r-1}{r} = \frac{(n+r-1)!}{r!(n-1)!}\]

Step-by-Step Manual Combinatorics Calculation

Let us calculate both permutations and combinations for \(n = 6\) items, choosing \(r = 4\) without repetition.

Permutation calculation (Order matters):

Step 1: Write down the permutation formula: \(P(6, 4) = \frac{6!}{(6-4)!} = \frac{6!}{2!}\).

Step 2: Expand the factorials: \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\). \(2! = 2 \times 1 = 2\).

Step 3: Perform division: \(720 / 2 = 360\). There are 360 ordered permutations.

Step 4: Alternatively, simplify by expansion cancellation: \(P(6,4) = 6 \times 5 \times 4 \times 3 = 360\).

Combination calculation (Order does not matter):

Step 1: Write down the combination formula: \(C(6, 4) = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \times 2!}\).

Step 2: Simplify the fraction by cancelling out \(4!\) from the numerator and denominator: \[\frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{(4 \times 3 \times 2 \times 1) \times (2 \times 1)} = \frac{6 \times 5}{2 \times 1}\]

Step 3: Solve the remaining expression: \(30 / 2 = 15\). There are exactly 15 unordered combinations.

FAQs

Why are combination locks misnamed?

A combination lock requires dial numbers to be entered in a specific sequence (e.g., 25-10-05). If you enter the numbers in a different order (e.g., 10-05-25), the lock will not open. Because the order of numbers is critical, these devices are mathematically permutation locks, not combination locks.

What is the "Stars and Bars" theorem in combinations?

Stars and Bars is a graphical method used to solve combinations with repetition. Imagine distributing r identical objects (stars) into n distinct categories separated by dividers (bars). The number of ways to arrange the stars and bars is mathematically equivalent to picking r star positions from a total of r + n - 1 items, leading directly to the formula C(n+r-1, r).

What is the relationship between permutations and combinations?

For any set of parameters (n, r), the number of permutations is always equal to the number of combinations multiplied by r! (the number of ways to arrange the r selected items). Mathematically: P(n, r) = C(n, r) × r!. Therefore, permutations are always greater than or equal to combinations.

How do factorials relate to probability?

Factorials represent the number of possible outcomes in a sample space. In classical probability, the chance of a specific event occurring is the number of favorable arrangements divided by the total possible arrangements. Because these arrangements are calculated using permutations or combinations (which rely on factorials), factorials are foundational to probability theory.

Why does 0! equal 1?

Defining 0! = 1 is necessary to keep combinatorial formulas consistent. For example, if we choose n items from a set of n, there is exactly 1 way to do this. Using the combination formula: C(n, n) = n! / (n! × 0!) = 1 / 0!. For this fraction to equal 1, 0! must be defined as 1.

How the calculation works

1Input total items (n) and selections size (r).

2Toggle repetition rules checkbox to allow/disallow item reuse.

3Select calculation mode to choose the highlighted main output metric.

4The factorials card shows calculations for n!, r!, and (n-r)!.

5If n and r are 5 or less, the subsets panel generates and displays all exact subsets alphabetically.

Permutations vs Combinations Growth

Observe how permutations (ordered choices) scale much faster than combinations (unordered choices) as selection size increases.

How to understand and use this calculator

Foundations of Combinatorics: The Science of Selection and Arrangement

To solve these problems, combinatorics relies on two basic operations: permutations and combinations. The core distinction between them lies in one simple rule: order.

The Fundamental Counting Principle and Factorials

This leads directly to the concept of Factorials. The factorial of a positive integer \(n\) (written as \(n!\)) is the product of all positive integers less than or equal to \(n\):

\[n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1\]

Permutation Formulas: With and Without Repetition

We categorize permutations based on whether we are allowed to select the same item more than once (repetition):

1. Permutations Without Repetition:

You have \(n\) unique items and select \(r\) of them without replacement. The number of unique arrangements is given by:

\[P(n, r) = \frac{n!}{(n-r)!}\]

For instance, if you have 5 books and want to arrange 3 of them on a shelf, \(P(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60\) ways.

2. Permutations With Repetition:

Each time you make a choice from \(n\) items, you return the selected item to the pool. The number of arrangements is simply:

\[P_R(n, r) = n^r\]

This applies to lock codes or passwords. If a lock has a 3-digit dial and each dial has 10 digits (0-9), the number of permutations is \(10^3 = 1000\).

Combination Formulas: With and Without Repetition

Combinations are similarly classified by whether items can be repeated:

1. Combinations Without Repetition (Binomial Coefficient):

Selecting \(r\) items from \(n\) items without replacement, where order is irrelevant. The formula is:

\[C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}\]

2. Combinations With Repetition (Stars and Bars Theorem):

\[C_R(n, r) = \binom{n+r-1}{r} = \frac{(n+r-1)!}{r!(n-1)!}\]

Step-by-Step Manual Combinatorics Calculation

Let us calculate both permutations and combinations for \(n = 6\) items, choosing \(r = 4\) without repetition.

Permutation calculation (Order matters):

Step 1: Write down the permutation formula: \(P(6, 4) = \frac{6!}{(6-4)!} = \frac{6!}{2!}\).

Step 2: Expand the factorials: \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\). \(2! = 2 \times 1 = 2\).

Step 3: Perform division: \(720 / 2 = 360\). There are 360 ordered permutations.

Step 4: Alternatively, simplify by expansion cancellation: \(P(6,4) = 6 \times 5 \times 4 \times 3 = 360\).

Combination calculation (Order does not matter):

Step 1: Write down the combination formula: \(C(6, 4) = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \times 2!}\).

Step 3: Solve the remaining expression: \(30 / 2 = 15\). There are exactly 15 unordered combinations.

FAQs

Why are combination locks misnamed?

What is the "Stars and Bars" theorem in combinations?

What is the relationship between permutations and combinations?

How do factorials relate to probability?

Why does 0! equal 1?

Permutation & Combination Calculator

Save this combinatorics workspace

Inputs

Results

Permutations & Combinations Math

How the calculation works

Combinatorics Example scenario

Learn more about this calculation

Standard Deviation: Measuring Data Spread and Variability

Random Number Generation: True Random vs Pseudo-Random Numbers

Permutation & Combination Calculator

Save this combinatorics workspace

Inputs

Results

Key output metrics

Permutations vs Combinations Growth

How to understand and use this calculator

Foundations of Combinatorics: The Science of Selection and Arrangement

The Fundamental Counting Principle and Factorials

Permutation Formulas: With and Without Repetition

Combination Formulas: With and Without Repetition

Step-by-Step Manual Combinatorics Calculation

FAQs

Permutations & Combinations Math

How the calculation works

Combinatorics Example scenario

Learn more about this calculation

Standard Deviation: Measuring Data Spread and Variability

Random Number Generation: True Random vs Pseudo-Random Numbers

Key output metrics

Permutations vs Combinations Growth

How to understand and use this calculator

Foundations of Combinatorics: The Science of Selection and Arrangement

The Fundamental Counting Principle and Factorials

Permutation Formulas: With and Without Repetition

Combination Formulas: With and Without Repetition

Step-by-Step Manual Combinatorics Calculation

FAQs