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The factorial of a natural number n is the product of all natural numbers smaller than or equal to n.

On this page we provide a basic introduction to factorials and we explain how they are used in probability theory and statistics.

Table of Contents


The following is a formal definition.

Definition Let $nin U{2115} $. The factorial of n, denoted by $n!$, is:[eq1]

The expression $n!$ is read "n factorial".

This definition is extended to the number 0 by using the convention:[eq2]


For example, the factorial of 6 is[eq3]

It is frequent to encounter ratios of factorials, which can be computed by simplifying the common terms. For example,[eq4]

Use in probability

In the calculus of probabilities we often need to count permutations, combinations and partitions of objects. This can easily be done with factorials.

Counting permutations

A permutation is one of the possible ways of ordering n objects, from first to last.

The number of possible permutations is equal to $n!$.

Example Consider the first three letters of the alphabet: $a,b,c$. There are [eq5]ways of ordering these letters:[eq6]

Counting combinations

A combination is a way of selecting k objects from a list of n. The order of selection does not matter and each object can be selected only once.

The number of possible combinations is equal to[eq7]

Example The number of possible ways to choose a team of three people from a group of five is[eq8]

Counting partitions

A partition is a way of subdividing n objects into k groups having numerosities [eq9].

The number of possible partitions is[eq10]

Example The number of possible ways to assign six individuals to three teams of two people is[eq11]

Use in statistics

Factorials have numerous important applications in the analysis of probability distributions.

For example, they appear in the probability mass functions of:


The concept of factorial can be extended using the Gamma function[eq12]

Unlike the factorial, the Gamma function is defined also when $z$ is not an integer.

It has the property that[eq13]when n is an integer.

The Gamma function is often used in statistics, for example, in the probability density functions of:

In turn, the Gamma function is used to define the Beta function, which is found in the densities of

More details

An in-depth explanation of factorials can be found in the lecture entitled Permutations.

Keep reading the glossary

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Next entry: Heteroskedasticity

How to cite

Please cite as:

Taboga, Marco (2021). "Factorial", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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