Factorial

Selected factorials; values in scientific notation are rounded

$n$	$n!$
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5040
8	40320
9	362880
10	3628800
11	39916800
12	479001600
13	6227020800
14	87178291200
15	1307674368000
16	20922789888000
17	355687428096000
18	6402373705728000
19	121645100408832000
20	2432902008176640000
25	1.551121004×1025
50	3.041409320×1064
70	1.197857167×10100
100	9.332621544×10157
450	1.733368733×101000
1000	4.023872601×102567
3249	6.412337688×1010000
10000	2.846259681×1035659
25206	1.205703438×10100000
100000	2.824229408×10456573
205023	2.503898932×101000004
1000000	8.263931688×105565708
10100	1010101.9981097754820

In mathematics, the factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$:

$$n!=n\cdot (n-1)\cdot (n-2)\cdot (n-3)\cdot \cdots \cdot 3\cdot 2\cdot 1\,.$$

For example,

$$5!=5\cdot 4\cdot 3\cdot 2\cdot 1=120\,.$$

The value of 0! is 1, according to the convention for an empty product.

The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic use counts the possible distinct sequences – the permutations – of $n$ distinct objects: there are $n!$.

The factorial function can also be extended to non-integer arguments while retaining its most important properties by defining $x!=\Gamma (x+1)$, where $\Gamma$ is the gamma function; this is undefined when $x$ is a negative integer.

History

The concept of factorials has arisen independently in many cultures:

• In Indian mathematics, one of the earliest known descriptions of factorials comes from the Anuyogadvāra-sūtra, one of the canonical works of Jain literature, likely dating from 200 BCE to 100 CE. It separates out the sorted and reversed order of a set of items from the other ("mixed") orders, evaluating the number of mixed orders by subtracting two from the usual product formula for the factorial. The product rule for permutations was also described by 6th-century CE Jain monk Jinabhadra. Hindu scholars have been using factorial formulas since at least 1150, when Bhāskara II mentioned factorials in his work Līlāvatī, in connection with a problem of how many ways Vishnu could hold his four characteristic objects (a conch shell, discus, mace, and lotus flower) in his four hands, and a similar problem for a ten-handed god.
• In the mathematics of the Middle East, the Hebrew mystic book of creation Sefer Yetzirah, from the Talmudic period (200 to 500 CE), lists factorials up to 7! as part of an investigation into the number of words that can be formed from the Hebrew alphabet. Factorials were also studied for similar reasons by 8th-century Arab grammarian Al-Khalil ibn Ahmad al-Farahidi.
• In Europe, although Greek mathematics included some combinatorics, and Plato famously used 5040 (a factorial) as the population of an ideal community, in part because of its divisibility properties, there is no direct evidence of ancient Greek study of factorials. Instead, the first work on factorials in Europe was by Jewish scholars such as Shabbethai Donnolo, explicating the Sefer Yetzirah passage. In 1677, British author Fabian Stedman described the application of factorials to change ringing, a musical art involving the ringing of several tuned bells.

From the late 15th century onward, factorials became the subject of study by western mathematicians. In a 1494 treatise, Italian mathematician Luca Pacioli calculated factorials up to 11!, in connection with a problem of dining table arrangements. Christopher Clavius discussed factorials in a 1603 commentary on the work of Johannes de Sacrobosco, and in the 1640s, French polymath Marin Mersenne published large (but not entirely correct) tables of factorials, up to 64!, based on the work of Clavius. The power series for the exponential function, with the reciprocals of factorials for its coefficients, was first formulated in 1676 by Isaac Newton in a letter to Gottfried Wilhelm Leibniz. Other important works of early European mathematics on factorials include extensive coverage in a 1685 treatise by John Wallis, a study of their approximate values for large values of $n$ by Abraham de Moivre in 1721, a 1729 letter from James Stirling to de Moivre stating what became known as Stirling's approximation, and work at the same time by Daniel Bernoulli and Leonhard Euler formulating the continuous extension of the factorial function to the gamma function. Adrien-Marie Legendre included Legendre's formula, describing the exponents in the factorization of factorials into prime powers, in an 1808 text on number theory.

The notation $n!$ for factorials was introduced by the French mathematician Christian Kramp in 1808. Many other notations have also been used. Another later notation, in which the argument of the factorial was half-enclosed by the left and bottom sides of a box, was popular for some time in Britain and America but fell out of use, perhaps because it is difficult to typeset. The word "factorial" (originally French, factorielle) was first used in this sense in 1800 by Louis François Antoine Arbogast, in the first work on Faà di Bruno's formula.

Definition

The factorial function of a positive integer $n$ is defined by the product

$$n!=1\cdot 2\cdot 3\cdots (n-2)\cdot (n-1)\cdot n,$$

This may be written more concisely in product notation as

$$n!=\prod _{i=1}^{n}i.$$

In this product formula, all but the last term define a product of the same form, for a smaller factorial. This leads to a recurrence relation, according to which each value of the factorial function can be obtained by multiplying the previous value by $n$:

$$n!=(n-1)!\cdot n.$$

For example, $5!=4!\cdot 5=24\cdot 5=120$.

Factorial of zero

The factorial of $0$ is $1$, or in symbols, $0!=1$. There are several motivations for this definition:

• For $n=0$, the definition of $n!$ as a product involves the product of no numbers at all, and so is an example of the broader convention that the empty product, a product of no factors, is equal to the multiplicative identity.
• There is exactly one permutation of zero objects: with nothing to permute, the only rearrangement is to do nothing.
• This convention makes many identities in combinatorics valid for all valid choices of their parameters. For instance, the number of ways to choose all $n$ elements from a set of $n$ is ${\textstyle {\tbinom {n}{n}}={\tfrac {n!}{n!0!}}=1,}$ a binomial coefficient identity that would only be valid with $0!=1$.
• With $0!=1$, the recurrence relation for the factorial remains valid at $n=1$. Therefore, with this convention, a recursive computation of the factorial needs to have only the value for zero as a base case, simplifying the computation and avoiding the need for additional special cases.
• Setting $0!=1$ allows for the compact expression of many formulae, such as the exponential function, as a power series: ${\textstyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}$
• This choice matches the gamma function $0!=\Gamma (0+1)=1$, for which it is necessary in order to make the function continuous.

Applications

The earliest uses of the factorial function involve counting permutations: there are $n!$ different ways of arranging $n$ distinct objects into a sequence. Factorials appear more broadly in many formulas in combinatorics, to account for different orderings of objects. For instance the binomial coefficients ${\tbinom {n}{k}}$ count the $k$-element combinations (subsets of $k$ elements) from a set with $n$ elements, and can be computed from factorials using the formula

$${\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.$$

The Stirling numbers of the first kind sum to the factorials, and count the permutations of $n$ grouped into subsets with the same numbers of cycles. Another combinatorial application is in counting derangements, permutations that do not leave any element in its original position; the number of derangements of $n$ items is the nearest integer to $n!/e$.

In algebra, the factorials arise through the binomial theorem, which uses binomial coefficients to expand products of sums. They also occur in the coefficients used to relate certain families of polynomials to each other, for instance in Newton's identities for symmetric polynomials. Their use in counting permutations can also be restated algebraically: the factorials are the orders of finite symmetric groups. In calculus, factorials occur in Faà di Bruno's formula for chaining higher derivatives. In mathematical analysis, factorials frequently appear in the denominators of power series, most notably in the series for the exponential function,

$$e^{x}=1+{\frac {x}{1}}+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+\cdots =\sum _{i=0}^{\infty }{\frac {x^{i}}{i!}},$$

and in the coefficients of other Taylor series, where they cancel factors of $n!$ coming from the $n$th derivative of $x^{n}$. This usage of factorials in power series connects back to analytic combinatorics through the exponential generating function, which for a combinatorial class with $n_{i}$ elements of size $i$ is defined as the power series

$$\sum _{i=0}^{\infty }{\frac {x^{i}n_{i}}{i!}}.$$

In number theory, the most salient property of factorials is the divisibility of $n!$ by all positive integers up to $n$, described more precisely for prime factors by Legendre's formula. It follows that $n!\pm 1$ has only large prime factors, leading to a proof of Euclid's theorem that the number of primes is infinite. When $n!\pm 1$ is itself prime it is called a factorial prime; relatedly, Brocard's problem, also posed by Srinivasa Ramanujan, concerns the existence of square numbers of the form $n!+1$. In contrast, the numbers $n!+2,n!+3,\dots n!+n$ must all be composite, proving the existence of arbitrarily large prime gaps. An elementary proof of Bertrand's postulate on the existence of a prime in any interval of the form $[n,2n]$, one of the first results of Paul Erdős, was based on the divisibility properties of factorials. Factorials can be used in a primality test based on Wilson's theorem, which states that a number $p$ is prime if and only if

$$(p-1)!\equiv -1{\pmod {p}}.$$

The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials.

Factorials are used extensively in probability theory, for instance in the Poisson distribution and in the probabilities of random permutations. In computer science, beyond appearing in the analysis of brute-force searches over permutations, factorials arise in the lower bound of $\log _{2}n!$ on the number of comparisons needed to comparison sort a set of $n$ items, and in the analysis of chained hash tables, where the distribution of keys per cell can be accurately approximated by a Poisson distribution.

Properties

Growth and approximation

Comparison of the factorial, Stirling's approximation, and the simpler approximation $(n/e)^{n}$, on a doubly logarithmic scale

Relative error in a truncated Stirling series vs. number of terms

Main article: Stirling's approximation

As a function of $n$, the factorial has faster than exponential growth, but grows more slowly than a double exponential function. Its growth rate is similar to $n^{n}$, but slower by an exponential factor. One way of approaching this result is by taking the natural logarithm of the factorial, which turns its product formula into a sum, and then estimating the sum by an integral:

$$\ln n!=\sum _{x=1}^{n}\ln x\approx \int _{1}^{n}\ln x\,dx=n\ln n-n+1.$$

Exponentiating the result (and ignoring the negligible $+1$ term) approximates $n!$ as $(n/e)^{n}$. More carefully bounding the sum both above and below by an integral, using the trapezoid rule, shows that this estimate needs a correction term proportional to ${\sqrt {n}}$. The constant of proportionality for this correction can be found from the Wallis product, which expresses $\pi$ as a limiting ratio of factorials and powers of two. The result of these corrections is Stirling's approximation:

$$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\,.$$

Here, the $\sim$ symbol means that, as $n$ goes to infinity, the ratio between the left and right sides approaches one in the limit. Stirling's formula provides the first term in an asymptotic series that becomes even more accurate when taken to greater numbers of terms:

$$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+{\frac {1}{288n^{2}}}-{\frac {139}{51840n^{3}}}-{\frac {571}{2488320n^{4}}}+\cdots \right).$$

An alternative version uses only odd exponents in the correction terms:

$$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\exp \left({\frac {1}{12n}}-{\frac {1}{360n^{3}}}+{\frac {1}{1260n^{5}}}-{\frac {1}{1680n^{7}}}+\cdots \right).$$

Many other variations of these formulas have also been developed, by Srinivasa Ramanujan, Bill Gosper, and others.

The binary logarithm of the factorial, used to analyze comparison sorting, can be very accurately estimated using Stirling's approximation. In the formula below, the $O(1)$ term invokes big O notation.

$$\log _{2}n!=n\log _{2}n-(\log _{2}e)n+{\frac {1}{2}}\log _{2}n+O(1).$$

Divisibility

Main article: Legendre's formula

The product formula for the factorial implies that $n!$ is divisible by all prime numbers that are at most $n$, and by no larger prime numbers. More precise information about its divisibility is given by Legendre's formula, which gives the exponent of each prime $p$ in the prime factorization of $n!$ as

$$\sum _{i=1}^{\infty }\left\lfloor {\frac {n}{p^{i}}}\right\rfloor ={\frac {n-s_{p}(n)}{p-1}}.$$

Here $s_{p}(n)$ denotes the sum of the base-$p$ digits of $n$, and the exponent given by this formula can also be interpreted in advanced mathematics as the p-adic valuation of the factorial. Applying Legendre's formula to the product formula for binomial coefficients produces Kummer's theorem, a similar result on the exponent of each prime in the factorization of a binomial coefficient.

The special case of Legendre's formula for $p=5$ gives the number of trailing zeros in the decimal representation of the factorials. Legendre's formula implies that the exponent of the prime $p=2$ is always larger than the exponent for $p=5$, so each factor of five can be paired with a factor of two to produce one of these trailing zeros.

The product of two factorials, $m!\cdot n!$, always evenly divides $(m+n)!$. There are infinitely many factorials that equal the product of other factorials: if $n$ is itself any product of factorials, then $n!$ equals that same product multiplied by one more factorial, $(n-1)!$. The only known examples of factorials that are products of other factorials but are not of this "trivial" form are $9!=7!\cdot 3!\cdot 3!\cdot 2!$, $10!=7!\cdot 6!=7!\cdot 5!\cdot 3!$, and $16!=14!\cdot 5!\cdot 2!$. It would follow from the abc conjecture that there are only finitely many nontrivial examples.

The greatest common divisor of the values of a primitive polynomial over the integers must be a divisor of the factorial of the polynomial's degree.

Continuous interpolation and non-integer generalization

The gamma function provides a continuous interpolation of the factorial to non-integer values

Absolute values of the complex gamma function, showing poles at non-positive integers

Main article: Gamma function

There are infinitely many ways to extend the factorials to a continuous function. The most widely used of these is the gamma function, which can be defined for positive real numbers as the integral

$$\Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx.$$

The resulting function is related to the factorial of a non-negative integer $n$ by the equation

$$n!=\Gamma (n+1),$$

which can be used as a definition of the factorial for other numbers. At all values $x$ for which both $\Gamma (x)$ and $\Gamma (x-1)$ are defined, the gamma function obeys the functional equation

$$\Gamma (n)=(n-1)\Gamma (n-1),$$

generalizing the recurrence relation for the factorials.

The same integral converges more generally for any complex number $z$ whose real part is positive. It can be extended to the rest of the complex plane by solving for Euler's reflection formula

$$\Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin \pi z}}.$$

However, this fails to assign a value to the gamma function at the non-positive integers, because it would lead to a division by zero. The result of this extension process is an analytic function, the analytic continuation of the integral formula for the gamma function. It has a nonzero value at all complex numbers, except for the non-positive integers where it has simple poles. Correspondingly, this provides a definition for the factorial at all complex numbers other than the negative integers. One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by the Bohr–Mollerup theorem, which states that the gamma function (offset by one) is the only log-convex function on the positive real numbers that interpolates the factorials and obeys the same functional equation. A related uniqueness theorem of Helmut Wielandt states that the complex gamma function and its scalar multiples are the only holomorphic functions on the positive complex half-plane that obey the functional equation and remain bounded for complex numbers with real part between 1 and 2.

Other complex functions that interpolate the factorial values include Hadamard's gamma function, which is an entire function over all the complex numbers, including the non-positive integers. In the p-adic numbers, it is not possible to continuously interpolate the factorial function directly, because the factorials of large integers (a dense subset of the p-adics) converge to zero according to Legendre's formula, forcing any continuous function that is close to their values to be zero everywhere. Instead, the p-adic gamma function provides a continuous interpolation of a modified form of the factorial, omitting the factors in the factorial that are divisible by p.

The digamma function is the logarithmic derivative of the gamma function. Just as the gamma function provides a continuous interpolation of the factorials, offset by one, the digamma function provides a continuous interpolation of the harmonic numbers, offset by the Euler–Mascheroni constant.

Computation

If efficiency is not a concern, computing factorials is trivial from an algorithmic point of view: successively multiplying a variable initialized to $1$ by the integers up to $n$ will compute $n!$. The simplicity of this computation makes it suitable as an example for beginning programmers in the use of different programming styles, including iteration, recursion, dynamic programming, and functional programming. In the unit-cost random-access machine model of computation, in which each arithmetic operation is assumed to take constant time, any of these methods will compute $n!$ in time $O(n)$. However, when computing factorials using exact integer arithmetic, this model of computation is only suitable for small values of $n$, small enough to allow all numbers in the computation to fit into a machine word. The values 12! and 20! are the largest factorials that can be stored in, respectively, the 32-bit and 64-bit integers. Floating point representations allow larger factorials to be computed in this way, but produce an approximate rather than exact result. For the exact computation of factorials of larger numbers, the use of arbitrary-precision arithmetic software is necessary, and it is more appropriate to model the time to perform a multiplication as a function of the number of digits in the result.

The asymptotically best efficiency is obtained by computing n! from its prime factorization. As documented by Peter Borwein, prime factorization allows n! to be computed in time O(n(log n log log n)²), provided that a fast multiplication algorithm is used (for example, the Schönhage–Strassen algorithm).

Related sequences and functions

Main article: List of factorial and binomial topics

Several other integer sequences are similar to or related to the factorials:

Alternating factorial

The alternating factorial is the absolute value of the alternating sum of the first $n$ factorials, ${\textstyle \sum _{i=1}^{n}(-1)^{n-i}i!}$. These have mainly been studied in connection with their primality; only finitely many of them can be prime, but a complete list of primes of this form is not known.

Bhargava factorial

The Bhargava factorials are a family of integer sequences defined by Manjul Bhargava with similar number-theoretic properties to the factorials, including the factorials themselves as a special case.

Double factorial

The product of all the odd integers up to some odd positive integer $n$ is called the double factorial of $n$, and denoted by $n!!$. That is,

$$(2k-1)!!=\prod _{i=1}^{k}(2i-1)={\frac {(2k)!}{2^{k}k!}}.$$

For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945. Double factorials are used in trigonometric integrals, in expressions for the gamma function at half-integers and the volumes of hyperspheres, and in counting binary trees and perfect matchings.

Exponential factorial

Just as triangular numbers sum the numbers from $1$ to $n$, and factorials take their product, the exponential factorial exponentiates. The exponential factorial of $n$, denoted as $n\$$, is defined recursively as $n^{(n-1)\$}$, with the base case $0\$=1\$.$ For example,

$$4\$=4^{3^{2^{1}}}=262144.$$

These numbers grow much more quickly than regular factorials or even hyperfactorials.

Falling factorial

The notations $(x)_{n}$ or $x^{\underline {n}}$ are sometimes used to represent the product of the n integers counting up to and including x, equal to $x!/(x-n)!$. This is also known as a falling factorial or backward factorial, and the $(x)_{n}$ notation is a Pochhammer symbol. Falling factorials occur as coefficients in the higher derivatives of polynomials, and in the factorial moments of random variables. They count the number of different sequences of $n$ distinct items that can be drawn from a universe of $x$ items.

Jordan–Pólya numbers

The Jordan–Pólya numbers are the products of factorials, allowing repetitions. Every tree has a symmetry group whose number of symmetries is a Jordan–Pólya number, and every Jordan–Pólya number counts the symmetries of some tree.

Primorial

The primorial $n\#$ is the product of prime numbers less than or equal to $n$; this construction gives them some similar divisibility properties to factorials, but unlike factorials they are squarefree. As with the factorial primes $n!\pm 1$, researchers have studied primorial primes $n\#\pm 1$.

Subfactorial

The subfactorial yields the number of derangements of a set of $n$ objects. It is sometimes denoted $!n$, and equals the closest integer to $n!/e$.

External links

• OEIS sequence A000142 (Factorial numbers)
• "Factorial". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
• Weisstein, Eric W. "Factorial". MathWorld.