math calculator

Permutation & Combination Calculator

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

Formula Verified by Dr. Elena Rostova, PhD

Quick actions: Save inputs, view saved runs, print, share, or export results

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

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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.

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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

Arrangements versus Selections: Permutations and Combinations

Combinatorics is the branch of mathematics concerned with counting, arranging, and selecting objects from a set. The two primary counting concepts are permutations and combinations. The key distinction between them is whether the order of choice matters.

Permutations count ordered arrangements. If the order in which items are selected changes the outcome, you are working with permutations. A classic example is a lock combination (where the sequence 1-2-3 is different from 3-2-1) or ranking competitors in a race.

Combinations count unordered selections. If the order of selection does not change the outcome, you are working with combinations. A classic example is drawing cards from a deck (holding Ace-King is the same as holding King-Ace) or choosing a committee from a pool of candidates.

Formulas for Counting with and without Replacement

The formulas used to count permutations and combinations depend on whether items can be selected more than once (repetition or replacement).

Permutations without Replacement: Arranging \(r\) items from a set of \(n\) distinct items:

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

Permutations with Replacement: Selecting \(r\) items from a set of \(n\) items where items can be reused. Each choice has \(n\) possibilities:

\[n^r\]

Combinations without Replacement: Choosing \(r\) items from a set of \(n\) distinct items where order does not matter:

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

Combinations with Replacement: Choosing \(r\) items from a set of \(n\) items where order does not matter and items can be reused (stars-and-bars method):

\[\left(\!\left(\begin{array}{c}n\\r\end{array}\right)\!\right) = _nC_r(n+r-1, r) = \frac{(n+r-1)!}{r!(n-1)!}\]

The Significance of Factorials and Large-Scale Counting

The factorial symbol (\(!\)) represents the product of all positive integers descending from a number down to 1 (e.g., \(4! = 4 \times 3 \times 2 \times 1 = 24\)). In counting problems, factorials represent the total number of ways to arrange a set of distinct objects.

Factorial values grow extremely fast (faster than exponential growth). For example, while \(5! = 120\), \(10! = 3,628,800\), and \(20!\) is approximately \(2.43 \times 10^{18}\). This calculator uses double-precision floating-point arithmetic to evaluate large factorials, switching to scientific notation when values exceed normal integer limits. By dividing terms before fully evaluating the factorials, our solver prevents decimal overflow and computes ratios of massive factorials.

Related concepts

Factorial

The descending product of integers representing the number of ways to arrange a set of objects.

Binomial Theorem

The algebraic expansion of powers of a binomial, where combinations define the coefficients.

Stars and Bars (Combinatorics)

A method for solving combinations with repetition allowed.

Frequently Asked Questions

What is a permutation?+

A permutation is an arrangement of objects in a specific order. If the order changes, it is considered a different permutation.

What is a combination?+

A combination is a selection of objects where the order does not matter. The group {A, B} is the same as {B, A}.

Why does the combination formula divide by r!?+

Dividing by r! removes duplicate groups. Since a selected group of r items can be arranged in r! different ways, dividing by r! ensures each unique group is only counted once.

What is a factorial?+

A factorial (n!) is the product of all positive integers less than or equal to n, e.g. 5! = 5 * 4 * 3 * 2 * 1 = 120.

Why is 0! equal to 1?+

By definition, 0! = 1. This ensures that combinatorics formulas work correctly at boundary conditions, such as choosing n items from a set of n (nCn = n! / (n! * 0!) = 1).

When do I use permutations vs combinations?+

Use permutations when order is important (seating charts, race finishes, lock codes). Use combinations when order is irrelevant (drawing cards, picking teams, mixing ingredients).

What is replacement or repetition?+

Replacement means an item can be selected more than once (like picking numbers for a lottery where the balls are returned to the bin). No replacement means each item can only be selected once.

What is the stars and bars method?+

A combinatorics technique used to solve combinations with replacement by representing selections as stars separated by bars.

Can r be greater than n?+

Without replacement, r cannot be greater than n because you cannot choose more distinct items than exist in the set. With replacement, r can be any positive integer.

How does combinatorics relate to probability?+

Probability is calculated as favorable outcomes divided by total outcomes. Permutations and combinations are used to count these outcome spaces.

What is the binomial coefficient?+

The binomial coefficient is another term for combinations, written as (n over r), representing the coefficients in the expansion of (x + y)^n.

How does the calculator handle large numbers?+

It simplifies ratios of factorials before calculating them (by canceling out common terms in the numerator and denominator) to prevent decimal overflow.

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

Arrangements versus Selections: Permutations and Combinations

Formulas for Counting with and without Replacement

The formulas used to count permutations and combinations depend on whether items can be selected more than once (repetition or replacement).

Permutations without Replacement: Arranging \(r\) items from a set of \(n\) distinct items:

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

Permutations with Replacement: Selecting \(r\) items from a set of \(n\) items where items can be reused. Each choice has \(n\) possibilities:

\[n^r\]

Combinations without Replacement: Choosing \(r\) items from a set of \(n\) distinct items where order does not matter:

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

Combinations with Replacement: Choosing \(r\) items from a set of \(n\) items where order does not matter and items can be reused (stars-and-bars method):

\[\left(\!\left(\begin{array}{c}n\\r\end{array}\right)\!\right) = _nC_r(n+r-1, r) = \frac{(n+r-1)!}{r!(n-1)!}\]

The Significance of Factorials and Large-Scale Counting

Related concepts

Factorial

The descending product of integers representing the number of ways to arrange a set of objects.

Binomial Theorem

The algebraic expansion of powers of a binomial, where combinations define the coefficients.

Stars and Bars (Combinatorics)

A method for solving combinations with repetition allowed.

Frequently Asked Questions

What is a permutation?+

A permutation is an arrangement of objects in a specific order. If the order changes, it is considered a different permutation.

What is a combination?+

A combination is a selection of objects where the order does not matter. The group {A, B} is the same as {B, A}.

Why does the combination formula divide by r!?+

Dividing by r! removes duplicate groups. Since a selected group of r items can be arranged in r! different ways, dividing by r! ensures each unique group is only counted once.

What is a factorial?+

A factorial (n!) is the product of all positive integers less than or equal to n, e.g. 5! = 5 * 4 * 3 * 2 * 1 = 120.

Why is 0! equal to 1?+

By definition, 0! = 1. This ensures that combinatorics formulas work correctly at boundary conditions, such as choosing n items from a set of n (nCn = n! / (n! * 0!) = 1).

When do I use permutations vs combinations?+

Use permutations when order is important (seating charts, race finishes, lock codes). Use combinations when order is irrelevant (drawing cards, picking teams, mixing ingredients).

What is replacement or repetition?+

Replacement means an item can be selected more than once (like picking numbers for a lottery where the balls are returned to the bin). No replacement means each item can only be selected once.

What is the stars and bars method?+

A combinatorics technique used to solve combinations with replacement by representing selections as stars separated by bars.

Can r be greater than n?+

Without replacement, r cannot be greater than n because you cannot choose more distinct items than exist in the set. With replacement, r can be any positive integer.

How does combinatorics relate to probability?+

Probability is calculated as favorable outcomes divided by total outcomes. Permutations and combinations are used to count these outcome spaces.

What is the binomial coefficient?+

The binomial coefficient is another term for combinations, written as (n over r), representing the coefficients in the expansion of (x + y)^n.

How does the calculator handle large numbers?+

It simplifies ratios of factorials before calculating them (by canceling out common terms in the numerator and denominator) to prevent decimal overflow.

Permutation & Combination Calculator

Inputs

Results

Bookmark this Combinatorics Solver

Permutations & Combinations Math

How the calculation works

Combinatorics Example scenario

Learn more about this calculation

Standard Deviation Guide: Measure Data Spread Clearly

Random Number Generation Guide: Seeded PRNGs, True Randomness, and Safe Use Cases

Permutation & Combination Calculator

Inputs

Results

Bookmark this Combinatorics Solver

Key output metrics

Permutations vs Combinations Growth

How to understand and use this calculator

Arrangements versus Selections: Permutations and Combinations

Formulas for Counting with and without Replacement

The Significance of Factorials and Large-Scale Counting

Related concepts

Frequently Asked Questions

Permutations & Combinations Math

How the calculation works

Combinatorics Example scenario

Learn more about this calculation

Standard Deviation Guide: Measure Data Spread Clearly

Random Number Generation Guide: Seeded PRNGs, True Randomness, and Safe Use Cases

Key output metrics

Permutations vs Combinations Growth

How to understand and use this calculator

Arrangements versus Selections: Permutations and Combinations

Formulas for Counting with and without Replacement

The Significance of Factorials and Large-Scale Counting

Related concepts

Frequently Asked Questions