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- Statistics LibreTexts - Combinations and permutations (Nov. 07, 2026)
permutations and combinations
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What are permutations and combinations?
In mathematics, permutations and combinations both involve selecting subsets from a set of objects. However, permutations consider the order of selection, while combinations do not.
- Permutation A permutation is an ordered arrangement of a set of objects. The number of permutations of (n) things taken (k) at a time is denoted as ( _nP_k = \frac{n!}{(n - k)!} ).
- Combination A combination is the selection of (k) objects from a set of (n) objects where order doesn't matter. The number of such subsets is denoted by ( _nC_k = \frac{n!}{k!(n - k)!} ).
In essence, permutations count how many ways you can arrange items, while combinations count how many ways you can choose items.
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What is a permutation in mathematics?
In mathematics, a permutation is a way to select objects from a set to form subsets, where the order of selection matters. Specifically, a permutation is an ordered arrangement of a set of objects.
If you have n objects and want to form permutations using k of them, the number of different permutations possible is denoted by the symbol (_nP_k). The formula to calculate this is:
[_nP_k = \frac{n!}{(n - k)!}]
where "!" (factorial) means multiplying all consecutive positive integers from 1 up to n, and 0! is defined as 1.
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What is a combination in mathematics?
In mathematics, combinations refer to the selection of subsets from a set of objects where the order of selection does not matter.
To illustrate the concept, consider selecting a pair of letters from a set of five distinguishable letters (A, B, C, D, and E). In combinations, AB is the same as BA, so only one of these is counted. Therefore, there are only 10 possible subsets: AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE.
The number of such subsets is written as ({n}C{k}), read as "n choose k." The formula for combinations is:
[
{n}C{k} = \frac{n!}{k!(n-k)!}
]This is the same as the (n, k) binomial coefficient. These combinations are sometimes called k-subsets.
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How do permutations differ from combinations?
In mathematics, permutations and combinations are ways to select subsets from a set of objects.
- Permutations consider the order of selection. For example, given the letters A, B, C, D, and E, the selections AB and BA are distinct permutations. The number of permutations of n objects taken k at a time is denoted by ( _nP_k ) with the formula ( _nP_k = \frac{n!}{(n - k)!} ).
- Combinations, on the other hand, do not consider the order of selection. Using the same example, AB and BA are considered the same combination. The number of combinations of n objects taken k at a time is denoted by ( _nC_k ) with the formula ( _nC_k = \frac{n!}{k!(n - k)!} ).
In essence, permutations are about arrangements, while combinations are about selections.
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What are the basic formulas for calculating permutations and combinations?
Permutations determine the number of ways to arrange a set of n objects taken k at a time, considering the order of selection. The formula for permutations is expressed as:
[ _nP_k = \frac{n!}{(n - k)!} ]
where "!" denotes the factorial operation.
Combinations calculate the number of ways to choose k objects from a set of n objects without regard to order. The formula for combinations is given by:
[ _nC_k = \frac{n!}{k!(n - k)!} ]
Combinations are equivalent to binomial coefficients.
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How are permutations and combinations used to solve real-life problems?
Permutations and combinations are used to calculate the number of possible outcomes in different situations. Permutations consider the order of selection, while combinations do not.
Examples:
* Figuring out all possible arrangements of letters in a word is a permutation problem.
* Determining the number of ways to choose a team of players from a larger group is a combination problem.
* Probability theory: Determining the chance of an event occurring multiple times in independent trials.
* Combinations can be used to count the number of distinct solutions of equations.2 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor. By considering the ratio of the number of desired subsets to the number of all possible subsets for many games of chance in the 17th century, the French mathematicians Blaise Pascal and Pierre de Fermat gave impetus to the development of combinatorics and probability theory.
The concepts of and differences between permutations and combinations can be illustrated by examination of all the different ways in which a pair of objects can be selected from five distinguishable objects—such as the letters A, B, C, D, and E. If both the letters selected and the order of selection are considered, then the following 20 outcomes are possible:
Each of these 20 different possible selections is called a permutation. In particular, they are called the permutations of five objects taken two at a time, and the number of such permutations possible is denoted by the symbol 5P2, read “5 permute 2.” In general, if there are n objects available from which to select, and permutations (P) are to be formed using k of the objects at a time, the number of different permutations possible is denoted by the symbol nPk. A formula for its evaluation is nPk = n!/(n − k)! The expression n!—read “n factorial”—indicates that all the consecutive positive integers from 1 up to and including n are to be multiplied together, and 0! is defined to equal 1. For example, using this formula, the number of permutations of five objects taken two at a time is
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(For k = n, nPk = n! Thus, for 5 objects there are 5! = 120 arrangements.)
For combinations, k objects are selected from a set of n objects to produce subsets without ordering. Contrasting the previous permutation example with the corresponding combination, the AB and BA subsets are no longer distinct selections; by eliminating such cases there remain only 10 different possible subsets—AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE.
The number of such subsets is denoted by nCk, read “n choose k.” For combinations, since k objects have k! arrangements, there are k! indistinguishable permutations for each choice of k objects; hence dividing the permutation formula by k! yields the following combination formula:
This is the same as the (n, k) binomial coefficient (see binomial theorem; these combinations are sometimes called k-subsets). For example, the number of combinations of five objects taken two at a time is
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The formulas for nPk and nCk are called counting formulas since they can be used to count the number of possible permutations or combinations in a given situation without having to list them all.