View source for Factorial

{{Short description|Product of numbers from 1 to n}}
{| class="wikitable" style="margin:0 0 0 1em; text-align:right; float:right;"
|+ Selected factorials; values in scientific notation are rounded
|-
! <math>n</math>
! <math>n!</math>
|-
| 0 || 1
|-
| 1 || 1
|-
| 2 || 2
|-
| 3 || 6
|-
| 4 || 24
|-
| 5 || 120
|-
| 6 || 720
|-
| 7 || {{val|5040|fmt=gaps}}
|-
| 8 || {{val|40320}}
|-
| 9 || {{val|362880}}
|-
| 10 || {{val|3628800}}
|-
| 11 || {{val|39916800}}
|-
| 12 || {{val|479001600}}
|-
| 13 || {{val|6227020800}}
|-
| 14 || {{val|87178291200}}
|-
| 15 || {{val|1307674368000}}
|-
| 16 || {{val|20922789888000}}
|-
| 17 || {{val|355687428096000}}
|-
| 18 || {{val|6402373705728000}}
|-
| 19 || {{val|121645100408832000}}
|-
| 20 || {{val|2432902008176640000}}
|-
| 25 
| style="text-align:left" | {{val|1.551121004|e=25}}
|-
| 50 
| style="text-align:left" | {{val|3.041409320|e=64}}
|-
| 70 
| style="text-align:left" | {{val|1.197857167|e=100}}
|-
| 100 
| style="text-align:left" | {{val|9.332621544|e=157}}
|-
| 450 
| style="text-align:left" | {{val|1.733368733|e=1000|fmt=gaps}}
|-
| {{val|1000|fmt=gaps}} 
| style="text-align:left" | {{val|4.023872601|e=2567|fmt=gaps}}
|-
| {{val|3249|fmt=gaps}} 
| style="text-align:left" | {{val|6.412337688|e=10000}}
|-
| {{val|10000|fmt=gaps}} 
| style="text-align:left" | {{val|2.846259681|e=35659}}
|-
| {{val|25206|fmt=gaps}} 
| style="text-align:left" | {{val|1.205703438|e=100000}}
|-
| {{val|100000|fmt=gaps}} 
| style="text-align:left" | {{val|2.824229408|e=456573}}
|-
| {{val|205023|fmt=gaps}} 
| style="text-align:left" | {{val|2.503898932|e=1000004}}
|-
| {{val|1000000|fmt=gaps}} 
| style="text-align:left" | {{val|8.263931688|e=5565708}}
|-
| [[Googol|{{val|e=100}}]] ||10<sup>{{val|e=101.9981097754820}}</sup>
|}

In [[Mathematics|mathematics]], the '''factorial''' of a non-negative {{nowrap|[[Integer|integer]] <math>n</math>,}} denoted {{nowrap|by <math>n!</math>,}} is the [[Product (mathematics)|product]] of all positive integers less than or equal {{nowrap|to <math>n</math>.}} The factorial {{nowrap|of <math>n</math>}} also equals the product of <math>n</math> with the next smaller factorial:
<math display=block>
\begin{align}
n! &= n  \times  (n-1)  \times (n-2)  \times  (n-3) \times \cdots \times  3 \times  2 \times  1 \\
   &= n\times(n-1)!\\
\end{align}</math>
For example,
<math display=block>5! = 5\times 4! = 5  \times  4  \times  3  \times  2  \times  1 = 120. </math>
The value of 0! is 1, according to the convention for an [[Empty product|empty product]].<ref name=gkp>{{cite book|first1=Ronald L.|last1=Graham|first2=Donald E.|last2=Knuth|first3=Oren|last3=Patashnik|date=1988|title=Concrete Mathematics|publisher=Addison-Wesley|location=Reading, MA|isbn=0-201-14236-8|title-link=Concrete Mathematics|page=111}}</ref>

Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of [[Religion:Jain literature|Jain literature]], and by Jewish mystics in the Talmudic book ''[[Unsolved:Sefer Yetzirah|Sefer Yetzirah]]''. The factorial operation is encountered in many areas of mathematics, notably in [[Combinatorics|combinatorics]], where its most basic use counts the possible distinct [[Sequence|sequence]]s – the [[Permutation|permutation]]s – of <math>n</math> distinct objects: there {{nowrap|are <math>n!</math>.}} In [[Mathematical analysis|mathematical analysis]], factorials are used in [[Power series|power series]] for the [[Exponential function|exponential function]] and other functions, and they also have applications in [[Algebra|algebra]], [[Number theory|number theory]], [[Probability theory|probability theory]], and [[Computer science|computer science]].

Much of the mathematics of the factorial function was developed beginning in the late 18th and early 19th centuries.
[[Stirling's approximation]] provides an accurate approximation to the factorial of large numbers, showing that it grows more quickly than [[Exponential growth|exponential growth]]. [[Legendre's formula]] describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials. [[Biography:Daniel Bernoulli|Daniel Bernoulli]] and [[Biography:Leonhard Euler|Leonhard Euler]] interpolated the factorial function to a continuous function of [[Complex number|complex number]]s, except at the negative integers, the (offset) [[Gamma function|gamma function]].

Many other notable functions and number sequences are closely related to the factorials, including the [[Binomial coefficient|binomial coefficient]]s, [[Double factorial|double factorial]]s, falling factorials, [[Primorial|primorial]]s, and subfactorials. Implementations of the factorial function are commonly used as an example of different [[Computer programming|computer programming]] styles, and are included in [[Engineering:Scientific calculator|scientific calculator]]s and scientific computing software libraries. Although directly computing large factorials using the product formula or recurrence is not efficient, faster algorithms are known, matching to within a constant factor the time for fast [[Multiplication algorithm|multiplication algorithm]]s for numbers with the same number of digits.

==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,<ref name=datta-singh/> one of the canonical works of [[Religion:Jain literature|Jain literature]], which has been assigned dates varying from 300 BCE to 400 CE.<ref>{{cite journal | last = Jadhav | first = Dipak | date = August 2021 | doi = 10.18732/hssa67 | journal = History of Science in South Asia | pages = 209–231 | publisher = University of Alberta Libraries | title = Jaina Thoughts on Unity Not Being a Number | volume = 9| s2cid = 238656716 | doi-access = free }}. See discussion of dating on p. 211.</ref> 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.<ref name=datta-singh>{{cite book | last1 = Datta | first1 = Bibhutibhusan |  last2 = Singh | first2 = Awadhesh Narayan | editor1-last = Kolachana | editor1-first = Aditya | editor2-last = Mahesh | editor2-first = K. | editor3-last = Ramasubramanian | editor3-first = K. | contribution = Use of permutations and combinations in India | doi = 10.1007/978-981-13-7326-8_18 | pages = 356–376 | publisher = Springer Singapore | series = Sources and Studies in the History of Mathematics and Physical Sciences | title = Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla | year = 2019| s2cid = 191141516 }}. Revised by K. S. Shukla from a paper in ''Indian Journal of History of Science'' 27 (3): 231–249, 1992, {{MR|1189487}}. See p. 363.</ref> Hindu scholars have been using factorial formulas since at least 1150, when [[Biography:Bhāskara II|Bhāskara II]] mentioned factorials in his work [[Religion:Līlāvatī|Līlāvatī]], in connection with a problem of how many ways [[Religion:Vishnu|Vishnu]] could hold his four characteristic objects (a [[Religion:Shankha|conch shell]], discus, [[Religion:Kaumodaki|mace]], and [[Religion:Sacred lotus in religious art|lotus flower]]) in his four hands, and a similar problem for a ten-handed god.<ref>{{Cite journal |last=Biggs |first=Norman L. |date=May 1979 |title=The roots of combinatorics |journal=Historia Mathematica |volume=6 |issue=2 |pages=109–136 |doi=10.1016/0315-0860(79)90074-0  |doi-access= | mr = 0530622 }}</ref>
*In the mathematics of the Middle East, the Hebrew mystic book of creation ''[[Unsolved:Sefer Yetzirah|Sefer Yetzirah]]'', from the Talmudic period (200 to 500&nbsp;CE), lists factorials up to 7! as part of an investigation into the number of words that can be formed from the [[Social:Hebrew alphabet|Hebrew alphabet]].<ref name=katz>{{cite journal | last = Katz | first = Victor J. |  date = June 1994 | issue = 2 | journal = For the Learning of Mathematics | jstor = 40248112 | pages = 26–30 | title = Ethnomathematics in the classroom | volume = 14}}</ref><ref>[https://en.wikisource.org/wiki/Sefer_Yetzirah#CHAPTER_IV Sefer Yetzirah at Wikisource], Chapter IV, Section 4</ref> Factorials were also studied for similar reasons by 8th-century Arab grammarian Al-Khalil ibn Ahmad al-Farahidi.<ref name=katz/> Arab mathematician [[Biography:Ibn al-Haytham|Ibn al-Haytham]] (also known as Alhazen, c. 965 – c. 1040) was the first to formulate [[Wilson's theorem]] connecting the factorials with the [[Prime number|prime number]]s.<ref>{{cite journal | last = Rashed | first = Roshdi | author-link = Roshdi Rashed | doi = 10.1007/BF00717654 | issue = 4 | journal = Archive for History of Exact Sciences | language = fr | mr = 595903 | pages = 305–321 | title = Ibn al-Haytham et le théorème de Wilson | volume = 22 | year = 1980| s2cid = 120885025 }}</ref>
*In Europe, although Greek mathematics included some combinatorics, and [[Biography:Plato|Plato]] famously used 5,040 (a factorial) as the population of an ideal community, in part because of its divisibility properties,<ref>{{cite journal | last = Acerbi | first = F. | doi = 10.1007/s00407-003-0067-0 | issue = 6 | journal = Archive for History of Exact Sciences | jstor = 41134173 | mr = 2004966 | pages = 465–502 | title = On the shoulders of Hipparchus: a reappraisal of ancient Greek combinatorics | volume = 57 | year = 2003| s2cid = 122758966 }}</ref> 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.<ref>{{cite book|editor1-last=Wilson|editor1-first=Robin|editor2-last=Watkins|editor2-first=John J.|title=Combinatorics: Ancient & Modern|publisher=[[Organization:Oxford University Press|Oxford University Press]]|date=2013|isbn=978-0-19-965659-2|first=Victor J.|last=Katz|contribution=Chapter 4: Jewish combinatorics|pages=109–121}} See p. 111.</ref> In 1677, British author [[Biography:Fabian Stedman|Fabian Stedman]] described the application of factorials to [[Change ringing|change ringing]], a musical art involving the ringing of several tuned bells.<ref>{{cite journal | last = Hunt | first = Katherine | date = May 2018 | doi = 10.1215/10829636-4403136 | issue = 2 | journal = Journal of Medieval and Early Modern Studies | pages = 387–412 | title = The Art of Changes: Bell-Ringing, Anagrams, and the Culture of Combination in Seventeenth-Century England | volume = 48| url = https://ueaeprints.uea.ac.uk/id/eprint/83227/1/Accepted_Mnauscript.pdf }}</ref><ref>{{cite book|last=Stedman|first=Fabian|author-link=Fabian Stedman|title=Campanalogia|year=1677|place=London|pages=6–9}}  The publisher is given as "W.S." who may have been William Smith, possibly acting as agent for the Society of College Youths, to which society the "Dedicatory" is addressed.</ref>

From the late 15th century onward, factorials became the subject of study by Western mathematicians. In a 1494 treatise, Italian mathematician [[Biography:Luca Pacioli|Luca Pacioli]] calculated factorials up to 11!, in connection with a problem of dining table arrangements.<ref>{{cite book|editor1-last=Wilson|editor1-first=Robin|editor2-last=Watkins|editor2-first=John J.|title=Combinatorics: Ancient & Modern|publisher=[[Organization:Oxford University Press|Oxford University Press]]|date=2013|isbn=978-0-19-965659-2|first=Eberhard|last=Knobloch|contribution=Chapter 5: Renaissance combinatorics|pages=123–145}}  See p. 126.</ref> [[Biography:Christopher Clavius|Christopher Clavius]] discussed factorials in a 1603 commentary on the work of [[Biography:Johannes de Sacrobosco|Johannes de Sacrobosco]], and in the 1640s, French polymath [[Biography:Marin Mersenne|Marin Mersenne]] published large (but not entirely correct) tables of factorials, up to 64!, based on the work of Clavius.{{sfn|Knobloch|2013|pages=130–133}} The [[Power series|power series]] for the [[Exponential function|exponential function]], with the reciprocals of factorials for its coefficients, was first formulated in 1676 by [[Biography:Isaac Newton|Isaac Newton]] in a letter to [[Biography:Gottfried Wilhelm Leibniz|Gottfried Wilhelm Leibniz]].<ref name=exponential-series>{{cite book | last1 = Ebbinghaus | first1 = H.-D. |  last2 = Hermes | first2 = H. |  last3 = Hirzebruch | first3 = F. |  last4 = Koecher | first4 = M. | author4-link = Max Koecher | last5 = Mainzer | first5 = K. | author5-link = Klaus Mainzer | last6 = Neukirch | first6 = J. | author6-link = Jürgen Neukirch | last7 = Prestel | first7 = A. | last8 = Remmert | first8 = R. | author8-link = Reinhold Remmert | doi = 10.1007/978-1-4612-1005-4 | isbn = 0-387-97202-1 | mr = 1066206 | page = 131 | publisher = Springer-Verlag | location = New York | series = Graduate Texts in Mathematics | title = Numbers | url = https://books.google.com/books?id=Z53SBwAAQBAJ&pg=PA131 | volume = 123 | year = 1990}}</ref> Other important works of early European mathematics on factorials include extensive coverage in a 1685 treatise by [[Biography:John Wallis|John Wallis]], a study of their approximate values for large values of <math>n</math> by [[Biography:Abraham de Moivre|Abraham de Moivre]] in 1721, a 1729 letter from [[Biography:James Stirling (mathematician)|James Stirling]] to de Moivre stating what became known as [[Stirling's approximation]], and work at the same time by [[Biography:Daniel Bernoulli|Daniel Bernoulli]] and [[Biography:Leonhard Euler|Leonhard Euler]] formulating the continuous extension of the factorial function to the [[Gamma function|gamma function]].<ref>{{cite journal | last = Dutka | first = Jacques | doi = 10.1007/BF00389433 | issue = 3 | journal = Archive for History of Exact Sciences | jstor = 41133918 | mr = 1171521 | pages = 225–249 | title = The early history of the factorial function | volume = 43 | year = 1991| s2cid = 122237769 }}</ref> [[Biography:Adrien-Marie Legendre|Adrien-Marie Legendre]] included [[Legendre's formula]], describing the exponents in the [[Integer factorization|factorization]] of factorials into [[Prime power|prime power]]s, in an 1808 text on [[Number theory|number theory]].<ref>{{cite book|first=Leonard E.|last=Dickson|author-link=Leonard Eugene Dickson|title=History of the Theory of Numbers|title-link=History of the Theory of Numbers|volume=1|publisher=Carnegie Institution of Washington|year=1919|contribution=Chapter IX: Divisibility of factorials and multinomial coefficients|pages=263–278|contribution-url=https://archive.org/details/historyoftheoryo01dick/page/262}} See in particular p. 263.</ref>

The notation <math>n!</math> for factorials was introduced by the French mathematician Christian Kramp in 1808.<ref name=cajori/> Many other notations have also been used. Another later notation <math>\vert\!\underline{\,n}</math>, 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.<ref name=cajori>{{cite book | last = Cajori | first = Florian |  contribution = 448–449. Factorial "{{mvar|n}}" | contribution-url = https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n93 | pages = 71–77 | publisher = The Open Court Publishing Company | title = A History of Mathematical Notations, Volume II: Notations Mainly in Higher Mathematics | title-link = A History of Mathematical Notations | year = 1929}}</ref> The word "factorial" (originally French: ''factorielle'') was first used in 1800 by Louis François Antoine Arbogast,<ref>{{cite web|url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/f/|title=Earliest Known Uses of Some of the Words of Mathematics (F)|work=[[MacTutor History of Mathematics archive]]|publisher=University of St Andrews|first=Jeff|last=Miller}}</ref> in the first work on [[Faà di Bruno's formula]],<ref name=craik>{{cite journal | last = Craik | first = Alex D. D. | doi = 10.1080/00029890.2005.11920176 | issue = 2 | journal = [[The American Mathematical Monthly]] | jstor = 30037410 | mr = 2121322 | pages = 119–130 | title = Prehistory of Faà di Bruno's formula | volume = 112 | year = 2005| s2cid = 45380805 }}</ref> but referring to a more general concept of products of [[Arithmetic progression|arithmetic progression]]s. The "factors" that this name refers to are the terms of the product formula for the factorial.<ref>{{cite book|title=Du calcul des dérivations|last=Arbogast|first=Louis François Antoine|author-link=Louis François Antoine Arbogast|publisher=L'imprimerie de Levrault, frères|location=Strasbourg|year=1800|pages=364–365|url=https://archive.org/details/ducalculdesdri00arbouoft/page/364|language=fr}}</ref>

==Definition==
The factorial function of a positive integer <math>n</math> is defined by the product of all positive integers not greater than <math>n</math><ref name=gkp/>
<math display=block>n!  = 1 \cdot 2 \cdot 3 \cdots (n-2) \cdot (n-1) \cdot n.</math>
This may be written more concisely in [[Multiplication#Capital pi notation|product notation]] as<ref name=gkp/>
<math display=block>n!  = \prod_{i = 1}^n i.</math>

If this product formula is changed to keep all but the last term, it would define a product of the same form, for a smaller factorial. This leads to a [[Recurrence relation|recurrence relation]], according to which each value of the factorial function can be obtained by multiplying the previous value {{nowrap|by <math>n</math>:<ref name=hamkins/>}}
<math display=block> n! = n\cdot (n-1)!.</math>
For example, {{nowrap|<math>5! = 5\cdot 4!=5\cdot 24=120</math>.}}

===Factorial of zero===
The factorial {{nowrap|of <math>0</math>}} {{nowrap|is <math>1</math>,}} or in symbols, {{nowrap|<math>0!=1</math>.}} There are several motivations for this definition:
* For {{nowrap|<math>n=0</math>,}} the definition of <math>n!</math> as a product involves the product of no numbers at all, and so is an example of the broader convention that the [[Empty product|empty product]], a product of no factors, is equal to the multiplicative identity.<ref>{{cite book|title=CRC Handbook of Engineering Tables|first=Richard C.|last=Dorf|publisher=CRC Press|year=2003|page=5-5|contribution=Factorials|contribution-url=https://books.google.com/books?id=TCLOBgAAQBAJ&pg=SA5-PA5|isbn=978-0-203-00922-2}}</ref>
* There is exactly one permutation of zero objects: with nothing to permute, the only rearrangement is to do nothing.<ref name=hamkins>{{cite book | last = Hamkins | first = Joel David |  isbn = 978-0-262-53979-1 | location = Cambridge, Massachusetts | mr = 4205951 | page = 50 | publisher = MIT Press | title = Proof and the Art of Mathematics | url = https://books.google.com/books?id=Ns_tDwAAQBAJ&pg=PA50 | year = 2020}}</ref>
* This convention makes many identities in [[Combinatorics|combinatorics]] valid for all valid choices of their parameters. For instance, the number of ways to choose all <math>n</math> elements from a set of <math>n</math> is <math display=inline>\tbinom{n}{n} = \tfrac{n!}{n!0!} = 1,</math> a [[Binomial coefficient|binomial coefficient]] identity that would only be valid {{nowrap|with <math>0!=1</math>.<ref>{{cite journal | last1 = Goldenberg | first1 = E. Paul | last2 = Carter | first2 = Cynthia J. | date = October 2017 | doi = 10.5951/mathteacher.111.2.0104 | issue = 2 | journal = The Mathematics Teacher | jstor = 10.5951/mathteacher.111.2.0104 | pages = 104–110 | title = A student asks about (&minus;5)! | volume = 111}}</ref>}}
* With {{nowrap|<math>0!=1</math>,}} the recurrence relation for the factorial remains valid {{nowrap|at <math>n=1</math>.}} Therefore, with this convention, a [[Recursion|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.<ref>{{cite conference | last1 = Haberman | first1 = Bruria | last2 = Averbuch | first2 = Haim | editor1-last = Caspersen | editor1-first = Michael E. | editor2-last = Joyce | editor2-first = Daniel T. | editor3-last = Goelman | editor3-first = Don | editor4-last = Utting | editor4-first = Ian | contribution = The case of base cases: Why are they so difficult to recognize? Student difficulties with recursion | doi = 10.1145/544414.544441 | pages = 84–88 | publisher = Association for Computing Machinery | title = Proceedings of the 7th Annual SIGCSE Conference on Innovation and Technology in Computer Science Education, ITiCSE 2002, Aarhus, Denmark, June 24-28, 2002 | year = 2002}}</ref>
* Setting <math>0!=1</math> allows for the compact expression of many formulae, such as the [[Exponential function|exponential function]], as a [[Power series|power series]]: {{nowrap|<math display=inline> e^x = \sum_{n = 0}^\infty \frac{x^n}{n!}.</math><ref name=exponential-series/>}}
* This choice matches the [[Gamma function|gamma function]] {{nowrap|<math>0! = \Gamma(0+1) = 1</math>,}} and the gamma function must have this value to be a [[Continuous function|continuous function]].<ref>{{cite book|title=Solved Problems in Analysis: As Applied to Gamma, Beta, Legendre and Bessel Functions|series=Dover Books on Mathematics|first1=Orin J.|last1=Farrell|first2=Bertram|last2=Ross|publisher=Courier Corporation|year=1971|isbn=978-0-486-78308-6|page=10|url=https://books.google.com/books?id=fXPDAgAAQBAJ&pg=PA10}}</ref>

==Applications==
The earliest uses of the factorial function involve counting permutations: there are <math>n!</math> different ways of arranging <math>n</math> distinct objects into a sequence.<ref name="ConwayGuy1998">{{Cite book |title=The Book of Numbers |last1=Conway |first1=John H. |last2=Guy |first2=Richard |year=1998 |publisher=Springer Science & Business Media |isbn=978-0-387-97993-9 |language=en |author-link2=Richard K. Guy |pages=55–56|contribution=Factorial numbers}}</ref> Factorials appear more broadly in many formulas in [[Combinatorics|combinatorics]], to account for different orderings of objects. For instance the [[Binomial coefficient|binomial coefficient]]s <math>\tbinom{n}{k}</math> count the {{nowrap|<math>k</math>-element}} [[Combination|combination]]s (subsets of {{nowrap|<math>k</math> elements)}} from a set with {{nowrap|<math>n</math> elements,}} and can be computed from factorials using the formula{{sfn|Graham|Knuth|Patashnik|1988|p=156}} <math display=block>\binom{n}{k}=\frac{n!}{k!(n-k)!}.</math> The [[Stirling numbers of the first kind]] sum to the factorials, and count the permutations {{nowrap|of <math>n</math>}} grouped into subsets with the same numbers of cycles.<ref>{{cite book | last = Riordan | first = John | author-link = John Riordan (mathematician) | mr = 0096594 | page = 76 | publisher = Chapman & Hall | series = Wiley Publications in Mathematical Statistics | title = An Introduction to Combinatorial Analysis | url = https://books.google.com/books?id=Sbb_AwAAQBAJ&pg=PA76 | year = 1958| isbn = 9781400854332 }}</ref> Another combinatorial application is in counting [[Derangement|derangement]]s, permutations that do not leave any element in its original position; the number of derangements of <math>n</math> items is the [[Rounding|nearest integer]] {{nowrap|to <math>n!/e</math>.{{sfn|Graham|Knuth|Patashnik|1988|p=195}}}}

In [[Algebra|algebra]], the factorials arise through the [[Binomial theorem|binomial theorem]], which uses binomial coefficients to expand powers of sums.{{sfn|Graham|Knuth|Patashnik|1988|p=162}} They also occur in the coefficients used to relate certain families of polynomials to each other, for instance in [[Newton's identities]] for [[Symmetric polynomial|symmetric polynomial]]s.<ref>{{cite journal | last = Randić | first = Milan | doi = 10.1007/BF01205340 | issue = 1 | journal = Journal of Mathematical Chemistry | mr = 895533 | pages = 145–152 | title = On the evaluation of the characteristic polynomial via symmetric function theory | volume = 1 | year = 1987| s2cid = 121752631 }}</ref> Their use in counting permutations can also be restated algebraically: the factorials are the orders of finite [[Symmetric group|symmetric group]]s.<ref>{{cite book|title=Groups and Characters|first=Victor E.|last=Hill|publisher=Chapman & Hall|year=2000|mr=1739394|isbn=978-1-351-44381-4|page=70|contribution=8.1 Proposition: Symmetric group {{math|''S''<sub>''n''</sub>}}|contribution-url=https://books.google.com/books?id=yjL3DwAAQBAJ&pg=PA70}}</ref> In [[Calculus|calculus]], factorials occur in [[Faà di Bruno's formula]] for chaining higher derivatives.<ref name=craik/> In [[Mathematical analysis|mathematical analysis]], factorials frequently appear in the denominators of [[Power series|power series]], most notably in the series for the [[Exponential function|exponential function]],<ref name=exponential-series/> <math display=block>e^x=1+\frac{x}{1}+\frac{x^2}{2}+\frac{x^3}{6}+\cdots=\sum_{i=0}^{\infty}\frac{x^i}{i!},</math>
and in the coefficients of other [[Taylor series]] (in particular those of the [[Trigonometric functions|trigonometric]] and [[Hyperbolic functions|hyperbolic functions]]), where they cancel factors of <math>n!</math> coming from the {{nowrap|<math>n</math>th derivative}} {{nowrap|of <math>x^n</math>.<ref>{{cite book|title=Complexity and Criticality|series=Advanced physics texts|volume=1|first1=Kim|last1=Christensen|first2=Nicholas R.|last2=Moloney|publisher=Imperial College Press|year=2005|isbn=978-1-86094-504-5|contribution=Appendix A: Taylor expansion|page=341|contribution-url=https://books.google.com/books?id=bAIM1_EoQu0C&pg=PA341}}</ref>}} This usage of factorials in power series connects back to [[Analytic combinatorics|analytic combinatorics]] through the exponential generating function, which for a [[Combinatorial class|combinatorial class]] with <math>n_i</math> elements of {{nowrap|size <math>i</math>}} is defined as the power series<ref>{{cite book | last = Wilf | first = Herbert S. |  edition = 3rd | isbn = 978-1-56881-279-3 | mr = 2172781 | page = 22 | publisher = A K Peters | location = Wellesley, Massachusetts | title = generatingfunctionology | url = https://books.google.com/books?id=XOPMBQAAQBAJ&pg=PA22 | year = 2006}}</ref> <math display=block>\sum_{i=0}^{\infty} \frac{x^i n_i}{i!}.</math>

In [[Number theory|number theory]], the most salient property of factorials is the divisibility of <math>n!</math> by all positive integers up {{nowrap|to <math>n</math>,}} described more precisely for prime factors by [[Legendre's formula]]. It follows that arbitrarily large [[Prime number|prime number]]s can be found as the prime factors of the numbers
<math>n!\pm 1</math>, leading to a proof of [[Euclid's theorem]] that the number of primes is infinite.<ref>{{cite book | last = Ore | first = Øystein |  location = New York | mr = 0026059 | page = 66 | publisher = McGraw-Hill | title = Number Theory and Its History | url = https://books.google.com/books?id=Sl_6BPp7S0AC&pg=PA66 | year = 1948| isbn = 9780486656205 }}</ref> When <math>n!\pm 1</math> is itself prime it is called a [[Factorial prime|factorial prime]];<ref name=caldwell-gallot>{{cite journal | last1 = Caldwell | first1 = Chris K. | last2 = Gallot | first2 = Yves | doi = 10.1090/S0025-5718-01-01315-1 | issue = 237 | journal = [[Mathematics of Computation]] | mr = 1863013 | pages = 441–448 | title = On the primality of <math>n!\pm1</math> and <math>2\times3\times5\times\dots\times p\pm1</math> | volume = 71 | year = 2002| doi-access = free }}</ref> relatedly, [[Brocard's problem]], also posed by [[Biography:Srinivasa Ramanujan|Srinivasa Ramanujan]], concerns the existence of [[Square number|square number]]s of the form {{nowrap|<math>n!+1</math>.<ref>{{cite book | last = Guy | first = Richard K. | author-link = Richard K. Guy | contribution = D25: Equations involving factorial <math>n</math> | doi = 10.1007/978-0-387-26677-0 | edition = 3rd | isbn = 0-387-20860-7 | mr = 2076335 | pages = 301–302 | publisher = Springer-Verlag | location = New York | series = Problem Books in Mathematics | title = Unsolved Problems in Number Theory | year = 2004| volume = 1 }}</ref>}} In contrast, the numbers <math>n!+2,n!+3,\dots n!+n</math> must all be composite, proving the existence of arbitrarily large [[Prime gap|prime gap]]s.<ref>{{cite book|title=Closing the Gap: The Quest to Understand Prime Numbers|first=Vicky|last=Neale|author-link=Vicky Neale|publisher=Oxford University Press|year=2017|isbn=978-0-19-878828-7|pages=146–147|url=https://books.google.com/books?id=T7Q1DwAAQBAJ&pg=PA146}}</ref> An elementary [[Proof of Bertrand's postulate|proof of Bertrand's postulate]] on the existence of a prime in any interval of the {{nowrap|form <math>[n,2n]</math>,}} one of the first results of [[Biography:Paul Erdős|Paul Erdős]], was based on the divisibility properties of factorials.<ref>{{cite journal | last = Erdős | first = Pál | author-link = Paul Erdős | journal = Acta Litt. Sci. Szeged | language = de | pages = 194–198 | title = Beweis eines Satzes von Tschebyschef | trans-title = Proof of a theorem of Chebyshev | url = https://users.renyi.hu/~p_erdos/1932-01.pdf | volume = 5 | year = 1932 | zbl = 0004.10103}}</ref><ref>{{cite book | last = Chvátal | first = Vašek | author-link = Václav Chvátal | contribution = 1.5: Erdős's proof of Bertrand's postulate | contribution-url = https://books.google.com/books?id=_gVDEAAAQBAJ&pg=PA7 | doi = 10.1017/9781108912181 | isbn = 978-1-108-83183-3 | mr = 4282416 | pages = 7–10 | publisher = Cambridge University Press | location = Cambridge, England | title = The Discrete Mathematical Charms of Paul Erdős: A Simple Introduction | year = 2021| s2cid = 242637862 }}</ref> The [[Factorial number system|factorial number system]] is a [[Mixed radix|mixed radix]] notation for numbers in which the place values of each digit are factorials.<ref>{{cite journal | last = Fraenkel | first = Aviezri S. | author-link = Aviezri Fraenkel | doi = 10.1080/00029890.1985.11971550 | issue = 2 | journal = [[The American Mathematical Monthly]] | jstor = 2322638 | mr = 777556 | pages = 105–114 | title = Systems of numeration | volume = 92 | year = 1985}}</ref>

Factorials are used extensively in [[Probability theory|probability theory]], for instance in the [[Poisson distribution]]<ref>{{cite book | last = Pitman | first = Jim | contribution = 3.5: The Poisson distribution | doi = 10.1007/978-1-4612-4374-8 | pages = 222–236 | publisher = Springer | location = New York | title = Probability | year = 1993| isbn = 978-0-387-94594-1 }}</ref> and in the probabilities of [[Random permutation|random permutation]]s.{{sfn|Pitman|1993|p=153}} In [[Computer science|computer science]], beyond appearing in the analysis of [[Brute-force search|brute-force search]]es over permutations,<ref>{{cite book|title=Algorithm Design|first1=Jon|last1=Kleinberg|first2=Éva|last2=Tardos|publisher=Addison-Wesley|year=2006|page=55}}</ref> factorials arise in the lower bound of <math>\log_2 n!=n\log_2n-O(n)</math> on the number of comparisons needed to [[Comparison sort|comparison sort]] a set of <math>n</math> items,<ref name=knuth-sorting/> and in the analysis of chained [[Hash table|hash table]]s, where the distribution of keys per cell can be accurately approximated by a Poisson distribution.<ref>{{cite book|title=Algorithms|edition=4th|publisher=Addison-Wesley|first1=Robert|last1=Sedgewick|author1-link=Robert Sedgewick (computer scientist)|first2=Kevin|last2=Wayne|year=2011|isbn=978-0-13-276256-4|page=466|url=https://books.google.com/books?id=idUdqdDXqnAC&pg=PA466}}</ref> Moreover, factorials naturally appear in formulae from [[Physics:Quantum mechanics|quantum]] and [[Physics:Statistical physics|statistical physics]], where one often considers all the possible permutations of a set of particles. In [[Physics:Statistical mechanics|statistical mechanics]], calculations of [[Physics:Entropy|entropy]] such as [[Physics:Boltzmann's entropy formula|Boltzmann's entropy formula]] or the [[Physics:Sackur–Tetrode equation|Sackur–Tetrode equation]] must correct the count of [[Physics:Microstate (statistical mechanics)|microstates]] by dividing by the factorials of the numbers of each type of [[Identical particles|indistinguishable particle]] to avoid the [[Physics:Gibbs paradox|Gibbs paradox]]. Quantum physics provides the underlying reason for why these corrections are necessary.<ref>{{cite book|first=Mehran |last=Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher={{wipe|Cambridge University Press}} |isbn=978-0-521-87342-0 |oclc=860391091 |pages=107–110, 181–184}}</ref>

==Properties==
===Growth and approximation===
[[File:Mplwp factorial stirling loglog2.svg|thumb|Comparison of the factorial, Stirling's approximation, and the simpler approximation {{nowrap|<math>(n/e)^n</math>,}} on a doubly logarithmic scale]]
[[File:Stirling series relative error.svg|thumb|upright=1.6|Relative error in a truncated Stirling series vs. number of terms]]
{{main|Stirling's approximation}}
As a function {{nowrap|of <math>n</math>,}} the factorial has faster than [[Exponential growth|exponential growth]], but grows more slowly than a [[Double exponential function|double exponential function]].<ref>{{cite book | last = Cameron | first = Peter J. |  contribution = 2.4: Orders of magnitude | isbn = 978-0-521-45133-8 | pages = 12–14 | publisher = Cambridge University Press | title = Combinatorics: Topics, Techniques, Algorithms | year = 1994}}</ref> Its growth rate is similar {{nowrap|to <math>n^n</math>,}} but slower by an exponential factor. One way of approaching this result is by taking the [[Natural logarithm|natural logarithm]] of the factorial, which turns its product formula into a sum, and then estimating the sum by an integral:
<math display=block>\ln n! = \sum_{x=1}^n \ln x \approx \int_1^n\ln x\, dx=n\ln n-n+1.</math>
Exponentiating the result (and ignoring the negligible <math>+1</math> term) approximates <math>n!</math> as {{nowrap|<math>(n/e)^n</math>.<ref>{{cite book | last = Magnus | first = Robert | contribution = 11.10: Stirling's approximation | contribution-url = https://books.google.com/books?id=5hvxDwAAQBAJ&pg=PA391 | doi = 10.1007/978-3-030-46321-2 | isbn = 978-3-030-46321-2 | location = Cham | mr = 4178171 | page = 391 | publisher = Springer | series = Springer Undergraduate Mathematics Series | title = Fundamental Mathematical Analysis | year = 2020| s2cid = 226465639 }}</ref>}}
More carefully bounding the sum both above and below by an integral, using the trapezoid rule, shows that this estimate needs a correction factor proportional {{nowrap|to <math>\sqrt n</math>.}} The constant of proportionality for this correction can be found from the [[Wallis product]], which expresses <math>\pi</math> as a limiting ratio of factorials and powers of two. The result of these corrections is [[Stirling's approximation]]:<ref>{{cite book | last = Palmer | first = Edgar M. | contribution = Appendix II: Stirling's formula | isbn = 0-471-81577-2 | location = Chichester | mr = 795795 | pages = 127–128 | publisher = John Wiley & Sons | series = Wiley-Interscience Series in Discrete Mathematics | title = Graphical Evolution: An introduction to the theory of random graphs | year = 1985}}</ref>
<math display=block>n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\,.</math>
Here, the <math>\sim</math> symbol means that, as <math>n</math> goes to infinity, the ratio between the left and right sides approaches one in the [[Limit (mathematics)|limit]].
Stirling's formula provides the first term in an asymptotic series that becomes even more accurate when taken to greater numbers of terms:<ref name=asymptotic>{{cite journal | last1 = Chen | first1 = Chao-Ping | last2 = Lin | first2 = Long | doi = 10.1016/j.aml.2012.06.025 | issue = 12 | journal = Applied Mathematics Letters | mr = 2967837 | pages = 2322–2326 | title = Remarks on asymptotic expansions for the gamma function | volume = 25 | year = 2012| doi-access = free }}</ref>
<math display=block>
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).</math>
An alternative version uses only odd exponents in the correction terms:<ref name=asymptotic/>
<math display=block>
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).</math>
Many other variations of these formulas have also been developed, by [[Biography:Srinivasa Ramanujan|Srinivasa Ramanujan]], [[Biography:Bill Gosper|Bill Gosper]], and others.<ref name=asymptotic/>

The [[Binary logarithm|binary logarithm]] of the factorial, used to analyze [[Comparison sort|comparison sort]]ing, can be very accurately estimated using Stirling's approximation. In the formula below, the <math>O(1)</math> term invokes [[Big O notation|big O notation]].<ref name=knuth-sorting>{{cite book|title=The Art of Computer Programming, Volume 3: Sorting and Searching|first=Donald E.|last=Knuth|edition=2nd|publisher=Addison-Wesley|year=1998|isbn=978-0-321-63578-5|page=182|url=https://books.google.com/books?id=cYULBAAAQBAJ&pg=PA182}}</ref>
<math display=block>\log_2 n! = n\log_2 n-(\log_2 e)n + \frac12\log_2 n + O(1).</math>

===Divisibility and digits===
{{main|Legendre's formula}}
The product formula for the factorial implies that <math>n!</math> is divisible by all [[Prime number|prime number]]s that are at {{nowrap|most <math>n</math>,}} and by no larger prime numbers.<ref name=beiler>{{cite book|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|series=Dover Recreational Math Series|first=Albert H.|last=Beiler|publisher=Courier Corporation|year=1966|edition=2nd|isbn=978-0-486-21096-4|page=49|url=https://books.google.com/books?id=NbbbL9gMJ88C&pg=PA49}}</ref> More precise information about its divisibility is given by [[Legendre's formula]], which gives the exponent of each prime <math>p</math> in the prime factorization of <math>n!</math> as<ref>{{harvnb|Chvátal|2021}}. "1.4: Legendre's formula". pp. 6–7.</ref><ref name=padic>{{cite book | last = Robert | first = Alain M. |  contribution = 3.1: The {{nowrap|<math>p</math>-adic}} valuation of a factorial | doi = 10.1007/978-1-4757-3254-2 | isbn = 0-387-98669-3 | mr = 1760253 | pages = 241–242 | publisher = Springer-Verlag | location = New York | series = [[Graduate Texts in Mathematics]] | title = A Course in {{nowrap|<math>p</math>-adic}} Analysis | volume = 198 | year = 2000}}</ref>
<math display=block>\sum_{i=1}^\infty \left \lfloor \frac n {p^i} \right \rfloor=\frac{n - s_p(n)}{p - 1}.</math>
Here <math>s_p(n)</math> denotes the sum of the {{nowrap|[[Radix|base]]-<math>p</math>}} digits {{nowrap|of <math>n</math>,}} and the exponent given by this formula can also be interpreted in advanced mathematics as the [[P-adic valuation|{{mvar|p}}-adic valuation]] of the factorial.<ref name=padic/> Applying Legendre's formula to the product formula for [[Binomial coefficient|binomial coefficient]]s produces [[Kummer's theorem]], a similar result on the exponent of each prime in the factorization of a binomial coefficient.<ref>{{cite book | last1 = Peitgen |  first1 = Heinz-Otto | last2 = Jürgens | first2 = Hartmut |  last3 = Saupe | first3 = Dietmar |  contribution = Kummer's result and Legendre's identity | doi = 10.1007/b97624 | location = New York | pages = 399–400 | publisher = Springer | title = Chaos and Fractals: New Frontiers of Science | year = 2004| isbn=978-1-4684-9396-2 }}</ref> Grouping the prime factors of the factorial into [[Prime power|prime power]]s in different ways produces the [[Multiplicative partitions of factorials|multiplicative partitions of factorials]].<ref>{{Cite journal|last1=Alladi|first1=Krishnaswami|last2=Grinstead|first2=Charles|title=On the decomposition of n! into prime powers|journal=[[Journal of Number Theory]]|year=1977 |language=en|volume=9|issue=4|pages=452–458|doi=10.1016/0022-314x(77)90006-3|doi-access=free}}</ref>

The special case of Legendre's formula for <math>p=5</math> gives the number of [[Trailing zero#Factorial|trailing zeros]] in the decimal representation of the factorials.<ref name=koshy/> According to this formula, the number of zeros can be obtained by subtracting the base-5 digits of <math>n</math> from <math>n</math>, and dividing the result by four.<ref>{{cite OEIS|A027868|Number of trailing zeros in n!; highest power of 5 dividing n!}}</ref> Legendre's formula implies that the exponent of the prime <math>p=2</math> is always larger than the exponent for {{nowrap|<math>p=5</math>,}} so each factor of five can be paired with a factor of two to produce one of these trailing zeros.<ref name=koshy>{{cite book|title=Elementary Number Theory with Applications|first=Thomas|last=Koshy|edition=2nd|publisher=Elsevier|year=2007|isbn=978-0-08-054709-1|contribution=Example 3.12|page=178|contribution-url=https://books.google.com/books?id=d5Z5I3gnFh0C&pg=PA178}}</ref> The leading digits of the factorials are distributed according to [[Benford's law]].<ref>{{cite journal | last = Diaconis | first = Persi |  doi = 10.1214/aop/1176995891 | issue = 1 | journal = Annals of Probability | mr = 422186 | pages = 72–81 | title = The distribution of leading digits and uniform distribution mod 1 | volume = 5 | year = 1977| doi-access = free }}</ref> Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base.<ref>{{cite journal|last=Bird|first=R. S.|author-link=Richard Bird (computer scientist)|doi=10.1080/00029890.1972.11993051|journal=[[The American Mathematical Monthly]]|jstor=2978087|mr=302553|pages=367–370|title=Integers with given initial digits|volume=79|year=1972|issue=4}}</ref>

Another result on divisibility of factorials, [[Wilson's theorem]], states that <math>(n-1)!+1</math> is divisible by <math>n</math> if and only if <math>n</math> is a [[Prime number|prime number]].<ref name=beiler/> For any given {{nowrap|integer <math>x</math>,}} the [[Kempner function]] of <math>x</math> is given by the smallest <math>n</math> for which <math>x</math> divides {{nowrap|<math>n!</math>.<ref>{{cite journal | jstor = 2972639 | first = A. J. | last = Kempner | title = Miscellanea | journal = [[The American Mathematical Monthly]] | volume = 25 | pages = 201–210 | year = 1918 | doi = 10.2307/2972639 | issue = 5}}</ref>}} For almost all numbers (all but a subset of exceptions with asymptotic density zero), it coincides with the largest prime factor {{nowrap|of <math>x</math>.<ref>{{cite journal|title=The smallest factorial that is a multiple of {{mvar|n}} (solution to problem 6674)|journal=[[The American Mathematical Monthly]]|volume=101|year=1994|page=179|url=http://www-fourier.ujf-grenoble.fr/~marin/une_autre_crypto/articles_et_extraits_livres/irationalite/Erdos_P._Kastanas_I.The_smallest_factorial...-.pdf|first1=Paul|last1=Erdős|first2=Ilias|last2=Kastanas|doi=10.2307/2324376|jstor=2324376}}.</ref>}}

The product of two factorials, {{nowrap|<math>m!\cdot n!</math>,}} always evenly divides {{nowrap|<math>(m+n)!</math>.<ref name=bhargava/>}} There are infinitely many factorials that equal the product of other factorials: if <math>n</math> is itself any product of factorials, then <math>n!</math> equals that same product multiplied by one more factorial, {{nowrap|<math>(n-1)!</math>.}} The only known examples of factorials that are products of other factorials but are not of this "trivial" form are {{nowrap|<math>9!=7!\cdot 3!\cdot 3!\cdot 2!</math>,}} {{nowrap|<math>10!=7!\cdot 6!=7!\cdot 5!\cdot 3!</math>,}} and {{nowrap|<math>16!=14!\cdot 5!\cdot 2!</math>.<ref>{{harvnb|Guy|2004}}. "B23: Equal products of factorials". p. 123.</ref>}} It would follow from the [[Abc conjecture|{{mvar|abc}} conjecture]] that there are only finitely many nontrivial examples.<ref>{{cite journal | last = Luca | first = Florian |  doi = 10.1017/S0305004107000308 | issue = 3 | journal = [[Organization:Mathematical Proceedings of the Cambridge Philosophical Society|Mathematical Proceedings of the Cambridge Philosophical Society]] | mr = 2373957 | pages = 533–542 | title = On factorials which are products of factorials | volume = 143 | year = 2007| bibcode = 2007MPCPS.143..533L | s2cid = 120875316 }}</ref>

The greatest common divisor of the values of a [[Primitive part and content|primitive polynomial]] of degree <math>d</math> over the integers evenly divides {{nowrap|<math>d!</math>.<ref name=bhargava>{{cite journal | last = Bhargava | first = Manjul | author-link = Manjul Bhargava | url = https://www.maa.org/programs/maa-awards/writing-awards/the-factorial-function-and-generalizations | title = The factorial function and generalizations | journal = [[The American Mathematical Monthly]] | volume = 107 | year = 2000 | pages = 783–799 | doi = 10.2307/2695734 | issue = 9 | jstor = 2695734 | citeseerx = 10.1.1.585.2265
}}</ref>}}

===Continuous interpolation and non-integer generalization===
[[File:Generalized factorial function more infos.svg|thumb|upright=1.6|The gamma function (shifted one unit left to match the factorials) continuously interpolates the factorial to non-integer values]]
[[File:Gamma abs 3D.png|thumb|Absolute values of the complex gamma function, showing poles at non-positive integers]]
{{Main|Gamma function}}
There are infinitely many ways to extend the factorials to a [[Continuous function|continuous function]].<ref name=davis/> The most widely used of these<ref name=borwein-corless/> uses the [[Gamma function|gamma function]], which can be defined for positive real numbers as the [[Integral|integral]]
<math display=block> \Gamma(z) = \int_0^\infty x^{z-1} e^{-x}\,dx.</math>
The resulting function is related to the factorial of a non-negative integer <math>n</math> by the equation
<math display=block> n!=\Gamma(n+1),</math>
which can be used as a definition of the factorial for non-integer arguments.
At all values <math>x</math> for which both <math>\Gamma(x)</math> and <math>\Gamma(x-1)</math> are defined, the gamma function obeys the [[Functional equation|functional equation]]
<math display=block> \Gamma(n)=(n-1)\Gamma(n-1),</math>
generalizing the [[Recurrence relation|recurrence relation]] for the factorials.<ref name=davis>{{cite journal | last = Davis | first = Philip J. |  doi = 10.1080/00029890.1959.11989422 | journal = [[The American Mathematical Monthly]] | jstor = 2309786 | mr = 106810 | pages = 849–869 | title = Leonhard Euler's integral: A historical profile of the gamma function | url = https://www.maa.org/programs/maa-awards/writing-awards/leonhard-eulers-integral-an-historical-profile-of-the-gamma-function | volume = 66 | year = 1959| issue = 10 }}</ref>

The same integral converges more generally for any [[Complex number|complex number]] <math>z</math> whose real part is positive. It can be extended to the non-integer points in the rest of the [[Complex plane|complex plane]] by solving for Euler's [[Reflection formula|reflection formula]]
<math display=block>\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z}.</math>
However, this formula cannot be used at integers because, for them, the <math>\sin\pi z</math> term would produce a [[Division by zero|division by zero]]. The result of this extension process is an [[Analytic function|analytic function]], the [[Analytic continuation|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 [[Zeros and poles|simple poles]]. Correspondingly, this provides a definition for the factorial at all complex numbers other than the negative integers.<ref name=borwein-corless>{{cite journal | last1 = Borwein | first1 = Jonathan M. |  last2 = Corless | first2 = Robert M. | doi = 10.1080/00029890.2018.1420983 | issue = 5 | journal = [[The American Mathematical Monthly]] | mr = 3785875 | pages = 400–424 | title = Gamma and factorial in the ''Monthly'' | volume = 125 | year = 2018| arxiv = 1703.05349 | s2cid = 119324101 }}</ref>
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 [[Biography:Helmut Wielandt|Helmut Wielandt]] states that the complex gamma function and its scalar multiples are the only [[Holomorphic function|holomorphic function]]s on the positive complex half-plane that obey the functional equation and remain bounded for complex numbers with real part between 1 and 2.<ref>{{cite journal | last = Remmert | first = Reinhold |  doi = 10.1080/00029890.1996.12004726 | issue = 3 | journal = [[The American Mathematical Monthly]] | jstor = 2975370 | mr = 1376175 | pages = 214–220 | title = Wielandt's theorem about the {{nowrap|<math>\Gamma</math>-function}} | volume = 103 | year = 1996}}</ref>

Other complex functions that interpolate the factorial values include [[Hadamard's gamma function]], which is an [[Entire function|entire function]] over all the complex numbers, including the non-positive integers.<ref>{{cite book|first=J.|last=Hadamard|chapter=Sur l'expression du produit {{math|1·2·3· · · · ·(''n''−1)}} par une fonction entière|title=Œuvres de Jacques Hadamard|publisher=Centre National de la Recherche Scientifiques|location=Paris|date=1968|chapter-url=http://www.luschny.de/math/factorial/hadamard/HadamardFactorial.pdf|orig-date=1894|language=fr}}
</ref><ref>{{cite journal | last = Alzer | first = Horst | doi = 10.1007/s12188-008-0009-5 | issue = 1 | journal = Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | mr = 2541340 | pages = 11–23 | title = A superadditive property of Hadamard's gamma function | volume = 79 | year = 2009| s2cid = 123691692 }}</ref> In the [[P-adic number|{{mvar|p}}-adic number]]s, it is not possible to continuously interpolate the factorial function directly, because the factorials of large integers (a dense subset of the {{mvar|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|{{mvar|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 {{mvar|p}}.<ref>{{harvnb|Robert|2000}}. "7.1: The gamma function {{nowrap|<math>\Gamma_p</math>".}} pp. 366–385.</ref>

The [[Digamma function|digamma function]] is the [[Logarithmic derivative|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 number|harmonic number]]s, offset by the [[Euler–Mascheroni constant]].<ref>{{cite journal | last = Ross | first = Bertram | doi = 10.1080/0025570X.1978.11976704 | issue = 3 | journal = Mathematics Magazine | jstor = 2689999 | mr = 1572267 | pages = 176–179 | title = The psi function | volume = 51 | year = 1978}}</ref>

===Computation===
[[File:Vintage Texas Instruments Model SR-50A Handheld LED Electronic Calculator, Made in the USA, Price Was $109.50 in 1975 (8715012843).jpg|thumb|TI SR-50A, a 1975 calculator with a factorial key (third row, center right)]]
The factorial function is a common feature in [[Engineering:Scientific calculator|scientific calculator]]s.<ref>{{cite book|title=Understandable Statistics: Concepts and Methods|first1=Charles Henry|last1=Brase|first2=Corrinne Pellillo|last2=Brase|edition=11th|publisher=Cengage Learning|year=2014|isbn=978-1-305-14290-9|page=182|url=https://books.google.com/books?id=a8OiAgAAQBAJ&pg=PA182}}</ref> It is also included in scientific programming libraries such as the [[Python (programming language)|Python]] mathematical functions module<ref>{{cite web|url=https://docs.python.org/3/library/math.html|title=math — Mathematical functions|work=Python 3 Documentation: The Python Standard Library|access-date=2021-12-21}}</ref> and the [[Software:Boost (C++ libraries)|Boost C++ library]].<ref>{{cite web|url=https://www.boost.org/doc/libs/1_78_0/libs/math/doc/html/math_toolkit/factorials/sf_factorial.html| title=Factorial|work=Boost 1.78.0 Documentation: Math Special Functions|access-date=2021-12-21}}</ref> If efficiency is not a concern, computing factorials is trivial: just successively multiply a variable initialized {{nowrap|to <math>1</math>}} by the integers up {{nowrap|to <math>n</math>.}} The simplicity of this computation makes it a common example in the use of different computer programming styles and methods.<ref>{{cite book|title=Drawing Programs: The Theory and Practice of Schematic Functional Programming|first1=Tom|last1=Addis|first2=Jan|last2=Addis|publisher=Springer| year=2009| isbn=978-1-84882-618-2| pages=149–150|url=https://books.google.com/books?id=cWM7ZBfEl_0C&pg=PA149}}</ref>

The computation of <math>n!</math> can be expressed in [[Pseudocode|pseudocode]] using [[Iteration|iteration]]<ref>{{cite book|title=MATLAB Programming for Engineers|first=Stephen J.|last=Chapman|edition=6th|publisher=Cengage Learning|year=2019| isbn=978-0-357-03052-3| page=215|contribution=Example 5.2: The factorial function|contribution-url=https://books.google.com/books?id=jVEzEAAAQBAJ&pg=PA215}}</ref> as
 define factorial(''n''):
   ''f'' := 1
   for ''i'' := 1, 2, 3, ..., ''n'':
     ''f'' := ''f'' * ''i''
   return ''f''
or using [[Recursion (computer science)|recursion]]<ref>{{cite book|title=The Computing Universe: A Journey through a Revolution|first1=Tony|last1=Hey|first2=Gyuri|last2=Pápay|publisher=Cambridge University Press|year=2014|isbn=9781316123225|page=64|url=https://books.google.com/books?id=q4FIBQAAQBAJ&pg=PA64}}</ref> based on its recurrence relation as
 define factorial(''n''):
   if (''n'' = 0) return 1
   return ''n'' * factorial(''n'' − 1)
Other methods suitable for its computation include [[Memoization|memoization]],<ref>{{cite book|title=Hands-On Functional Programming with C++: An effective guide to writing accelerated functional code using C++17 and C++20| first=Alexandru|last=Bolboaca | publisher=Packt Publishing|year=2019|isbn=978-1-78980-921-3|page=188|url=https://books.google.com/books?id=GwSgDwAAQBAJ&pg=PA188}}</ref> [[Dynamic programming|dynamic programming]],<ref>{{cite book|title=Mastering Mathematica: Programming Methods and Applications| first=John W.|last=Gray|publisher=Academic Press|year=2014|isbn=978-1-4832-1403-0|pages=233–234| url=https://books.google.com/books?id=a4riBQAAQBAJ&pg=PA233}}</ref> and [[Functional programming|functional programming]].<ref>{{cite book|title=Scala From a Functional Programming Perspective: An Introduction to the Programming Language|volume=9980|series=Lecture Notes in Computer Science| first=Vicenç| last=Torra| publisher=Springer|year=2016|isbn=978-3-319-46481-7|page=96|url=https://books.google.com/books?id=eMwcDQAAQBAJ&pg=PA96}}</ref> The [[Computational complexity|computational complexity]] of these algorithms may be analyzed using the unit-cost [[Random-access machine|random-access machine]] model of computation, in which each arithmetic operation takes constant time and each number uses a constant amount of storage space. In this model, these methods can compute <math>n!</math> in time {{nowrap|<math>O(n)</math>,}} and the iterative version uses space {{nowrap|<math>O(1)</math>.}} Unless optimized for [[Tail recursion|tail recursion]], the recursive version takes linear space to store its [[Call stack|call stack]].<ref>{{cite book|title=Functional Programming and Its Applications: An Advanced Course| publisher=Cambridge University Press|series=CREST Advanced Courses|contribution=LISP, programming, and implementation| first=Gerald Jay|last=Sussman|year=1982|pages=29–72|isbn=978-0-521-24503-6}} See in particular [https://books.google.com/books?id=O_M8AAAAIAAJ&pg=PA34 p. 34].</ref> However, this model of computation is only suitable when <math>n</math> is small enough to allow <math>n!</math> to fit into a machine word.<ref>{{cite journal | last = Chaudhuri | first = Ranjan | date = June 2003 | doi = 10.1145/782941.782977 | issue = 2 | journal = ACM SIGCSE Bulletin | pages = 43–44 | publisher = Association for Computing Machinery | title = Do the arithmetic operations really execute in constant time? | volume = 35| s2cid = 13629142 }}</ref> The values 12! and 20! are the largest factorials that can be stored in, respectively, the [[32-bit computing|32-bit]]<ref name=fateman/> and [[64-bit computing|64-bit]] [[Integer (computer science)|integers]].<ref name=sigplan>{{cite journal | last1 = Winkler | first1 = Jürgen F. H. | last2 = Kauer | first2 = Stefan | date = March 1997 | doi = 10.1145/251634.251638 | issue = 3 | journal = ACM SIGPLAN Notices | pages = 38–41 | publisher = Association for Computing Machinery | title = Proving assertions is also useful | volume = 32| s2cid = 17347501 | doi-access = free }}</ref> Floating point can represent larger factorials, but approximately rather than exactly, and will still overflow for factorials larger than {{nowrap|<math>170!</math>.<ref name=fateman>{{cite web| url=http://people.eecs.berkeley.edu/~fateman/papers/factorial.pdf|title=Comments on Factorial Programs|date=April 11, 2006| publisher=University of California, Berkeley|first=Richard J.|last=Fateman|author-link=Richard Fateman}}</ref>}}

The exact computation of larger factorials involves [[Arbitrary-precision arithmetic|arbitrary-precision arithmetic]], because of [[Factorial#Growth and approximation|fast growth]] and [[Integer overflow|integer overflow]].  Time of computation can be analyzed as a function of the number of digits or bits in the result.<ref name=sigplan/> By Stirling's formula, <math>n!</math> has <math>b = O(n\log n)</math> bits.<ref name=borwein/> The [[Schönhage–Strassen algorithm]] can produce a {{nowrap|<math>b</math>-bit}} product in time {{nowrap|<math>O(b\log b\log\log b)</math>,}} and faster [[Multiplication algorithm|multiplication algorithm]]s taking time <math>O(b\log b)</math> are known.<ref>{{cite journal | last1 = Harvey | first1 = David | last2 = van der Hoeven | first2 = Joris |  doi = 10.4007/annals.2021.193.2.4 | issue = 2 | journal = [[Annals of Mathematics]] | mr = 4224716 | pages = 563–617 | series = Second Series | title = Integer multiplication in time <math>O(n \log n)</math>| volume = 193 | year = 2021| s2cid = 109934776 | url = https://hal.archives-ouvertes.fr/hal-02070778/file/nlogn.pdf }}</ref> However, computing the factorial involves repeated products, rather than a single multiplication, so these time bounds do not apply directly.  In this setting, computing <math>n!</math> by multiplying the numbers from 1 {{nowrap|to <math>n</math>}} in sequence is inefficient, because it involves <math>n</math> multiplications, a constant fraction of which take time <math>O(n\log^2 n)</math> each, giving total time {{nowrap|<math>O(n^2\log^2 n)</math>.}} A better approach is to perform the multiplications as a [[Divide-and-conquer algorithm|divide-and-conquer algorithm]] that multiplies a sequence of <math>i</math> numbers by splitting it into two subsequences of <math>i/2</math> numbers, multiplies each subsequence, and combines the results with one last multiplication. This approach to the factorial takes total time {{nowrap|<math>O(n\log^3 n)</math>:}} one logarithm comes from the number of bits in the factorial, a second comes from the multiplication algorithm, and a third comes from the divide and conquer.<ref>{{cite book|last=Arndt|first=Jörg| title=Matters Computational: Ideas, Algorithms, Source Code|publisher=Springer|year=2011|url=http://jjj.de/fxt/fxtbook.pdf| contribution=34.1.1.1: Computation of the factorial|pages=651–652}} See also "34.1.5: Performance", pp. 655–656.</ref>

Even better efficiency is obtained by computing {{math|''n''!}} from its prime factorization, based on the principle that [[Exponentiation by squaring|exponentiation by squaring]] is faster than expanding an exponent into a product.<ref name=borwein>{{cite journal | last = Borwein | first = Peter B. |  doi = 10.1016/0196-6774(85)90006-9 | issue = 3 | journal = Journal of Algorithms | mr = 800727 | pages = 376–380 | title = On the complexity of calculating factorials | volume = 6 | year = 1985}}</ref><ref name=schonhage>{{cite book|first=Arnold|last=Schönhage|year=1994|title=Fast algorithms: a multitape Turing machine implementation|publisher=B.I. Wissenschaftsverlag|page=226}}</ref> An algorithm for this by [[Biography:Arnold Schönhage|Arnold Schönhage]] begins by finding the list of the primes up {{nowrap|to <math>n</math>,}} for instance using the [[Sieve of Eratosthenes|sieve of Eratosthenes]], and uses Legendre's formula to compute the exponent for each prime. Then it computes the product of the prime powers with these exponents, using a recursive algorithm, as follows:
* Use divide and conquer to compute the product of the primes whose exponents are odd
* Divide all of the exponents by two (rounding down to an integer), recursively compute the product of the prime powers with these smaller exponents, and square the result
* Multiply together the results of the two previous steps
The product of all primes up to <math>n</math> is an <math>O(n)</math>-bit number, by the [[Prime number theorem|prime number theorem]], so the time for the first step is <math>O(n\log^2 n)</math>, with one logarithm coming from the divide and conquer and another coming from the multiplication algorithm. In the recursive calls to the algorithm, the prime number theorem can again be invoked to prove that the numbers of bits in the corresponding products decrease by a constant factor at each level of recursion, so the total time for these steps at all levels of recursion adds in a [[Geometric series|geometric series]] {{nowrap|to <math>O(n\log^2 n)</math>.}} The time for the squaring in the second step and the multiplication in the third step are again {{nowrap|<math>O(n\log^2 n)</math>,}} because each is a single multiplication of a number with <math>O(n\log n)</math> bits. Again, at each level of recursion the numbers involved have a constant fraction as many bits (because otherwise repeatedly squaring them would produce too large a final result) so again the amounts of time for these steps in the recursive calls add in a geometric series {{nowrap|to <math>O(n\log^2 n)</math>.}} Consequentially, the whole algorithm takes {{nowrap|time <math>O(n\log^2 n)</math>,}} proportional to a single multiplication with the same number of bits in its result.<ref name=schonhage/>

==Related sequences and functions==
{{main|List of factorial and binomial topics}}
Several other integer sequences are similar to or related to the factorials:

;Alternating factorial
:The [[Alternating factorial|alternating factorial]] is the absolute value of the alternating sum of the first <math>n</math> factorials, {{nowrap|<math display=inline>\sum_{i = 1}^n (-1)^{n - i}i!</math>.}} 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.<ref>{{harvnb|Guy|2004}}. "B43: Alternating sums of factorials". pp. 152–153.</ref>
;Bhargava factorial
:The [[Bhargava factorial]]s 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.<ref name=bhargava/>
;Double factorial
:The product of all the odd integers up to some odd positive {{nowrap|integer <math>n</math>}} is called the [[Double factorial|double factorial]] {{nowrap|of <math>n</math>,}} and denoted by {{nowrap|<math>n!!</math>.<ref name="callan">{{cite arXiv|title=A combinatorial survey of identities for the double factorial|first=David|last=Callan|eprint=0906.1317|year=2009|class=math.CO}}</ref>}} That is, <math display=block>(2k-1)!! = \prod_{i=1}^k (2i-1) = \frac{(2k)!}{2^k k!}.</math> For example, {{nowrap|1=9!! = 1 × 3 × 5  × 7 × 9 = 945}}. Double factorials are used in [[List of integrals of trigonometric functions|trigonometric integrals]],<ref>{{cite journal
 | last = Meserve | first = B. E.
 | doi = 10.2307/2306136
 | issue = 7
 | journal = [[The American Mathematical Monthly]]
 | mr = 1527019
 | pages = 425–426
 | title = Classroom Notes: Double Factorials
 | volume = 55
 | year = 1948| jstor = 2306136
 }}</ref> in expressions for the [[Gamma function|gamma function]] at [[Half-integer|half-integer]]s and the [[Volume of an n-ball|volumes of hyperspheres]],<ref>{{cite journal|title=Some dimension problems in molecular databases|first=Paul G.|last=Mezey|year=2009|journal=Journal of Mathematical Chemistry|volume=45|issue=1|pages=1–6|doi=10.1007/s10910-008-9365-8|s2cid=120103389}}.</ref> and in counting binary trees and [[Perfect matching|perfect matching]]s.<ref name="callan"/><ref>{{cite journal
 | last1 = Dale | first1 = M. R. T.
 | last2 = Moon | first2 = J. W.
 | doi = 10.1016/0378-3758(93)90035-5
 | issue = 1
 | journal = [[Journal of Statistical Planning and Inference]]
 | mr = 1209991
 | pages = 75–87
 | title = The permuted analogues of three Catalan sets
 | volume = 34
 | year = 1993}}.</ref>
;Exponential factorial
:Just as [[Triangular number|triangular number]]s sum the numbers from <math>1</math> {{nowrap|to <math>n</math>,}} and factorials take their product, the [[Exponential factorial|exponential factorial]] exponentiates. The exponential factorial is defined recursively {{nowrap|as <math>a_0 = 1,\ a_n = n^{a_{n - 1}}</math>.}} For example, the exponential factorial of 4 is <math display=block>4^{3^{2^{1}}}=262144.</math> These numbers grow much more quickly than regular factorials.<ref>{{cite journal | last1 = Luca | first1 = Florian |  last2 = Marques | first2 = Diego | issue = 3 | journal = [[Journal de Théorie des Nombres de Bordeaux]] | mr = 2769339 | pages = 703–718 | title = Perfect powers in the summatory function of the power tower | url = http://jtnb.cedram.org/item?id=JTNB_2010__22_3_703_0 | volume = 22 | year = 2010| doi = 10.5802/jtnb.740 | doi-access = free }}</ref>
;Falling factorial
:The notations <math>(x)_{n}</math> or <math>x^{\underline n}</math> are sometimes used to represent the product of the <math>n</math> integers counting up to and {{nowrap|including <math>x</math>,}} equal to {{nowrap|<math>x!/(x-n)!</math>.}} This is also known as a [[Falling and rising factorials|falling factorial]] or backward factorial, and the <math>(x)_{n}</math> notation is a Pochhammer symbol.{{sfn|Graham|Knuth|Patashnik|1988|pp=x, 47–48}} Falling factorials count the number of different sequences of <math>n</math> distinct items that can be drawn from a universe of <math>x</math> items.<ref>{{cite book | last = Sagan | first = Bruce E. |  contribution = Theorem 1.2.1 | contribution-url = https://books.google.com/books?id=DYgEEAAAQBAJ&pg=PA5 | isbn = 978-1-4704-6032-7 | location = Providence, Rhode Island | mr = 4249619 | page = 5 | publisher = American Mathematical Society | series = Graduate Studies in Mathematics | title = Combinatorics: the Art of Counting | volume = 210 | year = 2020}}</ref> They occur as coefficients in the higher derivatives of polynomials,<ref>{{cite book|first=G. H.|last=Hardy|author-link=G. H. Hardy|title=A Course of Pure Mathematics|title-link=A Course of Pure Mathematics|edition=3rd|publisher=Cambridge University Press|year=1921|contribution=Examples XLV|page=215|contribution-url=https://archive.org/details/coursepuremath00hardrich/page/n229}}</ref> and in the [[Factorial moment|factorial moment]]s of [[Random variable|random variable]]s.<ref>{{cite book | last1 = Daley | first1 = D. J. | last2 = Vere-Jones | first2 = D. | contribution = 5.2: Factorial moments, cumulants, and generating function relations for discrete distributions | contribution-url = https://books.google.com/books?id=Af7lBwAAQBAJ&pg=PA112 | isbn = 0-387-96666-8 | location = New York | mr = 950166 | page = 112 | publisher = Springer-Verlag | series = Springer Series in Statistics | title = An Introduction to the Theory of Point Processes | year = 1988}}</ref>
;Hyperfactorials
:The [[Hyperfactorial|hyperfactorial]] of <math>n</math> is the product <math>1^1\cdot 2^2\cdots n^n</math>. These numbers form the [[Discriminant|discriminant]]s of [[Hermite polynomials]].<ref>{{cite OEIS | 1=A002109 | 2=Hyperfactorials: Product_{k = 1..n} k^k}}</ref> They can be continuously interpolated by the [[K-function]],<ref>{{cite journal | last = Kinkelin | first = H. |  doi = 10.1515/crll.1860.57.122 | journal = [[Crelle's Journal|Journal für die reine und angewandte Mathematik]] | language = de | pages = 122–138 | title = Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechung | trans-title = On a transcendental variation of the gamma function and its application to the integral calculus | volume = 1860 | year = 1860| issue = 57 | s2cid = 120627417 }}</ref> and obey analogues to Stirling's formula<ref>{{cite journal | last = Glaisher | first = J. W. L. | author-link = James Whitbread Lee Glaisher | journal = [[Messenger of Mathematics]] | pages = 43–47 | title = On the product {{math|1<sup>1</sup>.2<sup>2</sup>.3<sup>3</sup>...''n''<sup>''n''</sup>}} | url = https://archive.org/details/messengermathem01glaigoog/page/n56 | volume = 7 | year = 1877}}</ref> and Wilson's theorem.<ref>{{cite journal | last1 = Aebi | first1 = Christian | last2 = Cairns | first2 = Grant | doi = 10.4169/amer.math.monthly.122.5.433 | issue = 5 | journal = [[The American Mathematical Monthly]] | jstor = 10.4169/amer.math.monthly.122.5.433 | mr = 3352802 | pages = 433–443 | title = Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials | volume = 122 | year = 2015| s2cid = 207521192 }}</ref>
;Jordan–Pólya numbers
:The [[Jordan–Pólya number]]s are the products of factorials, allowing repetitions. Every [[Tree (graph theory)|tree]] has a [[Symmetry group|symmetry group]] whose number of symmetries is a Jordan–Pólya number, and every Jordan–Pólya number counts the symmetries of some tree.<ref>{{cite OEIS|A001013|Jordan-Polya numbers: products of factorial numbers}}</ref>
;Primorial
:The [[Primorial|primorial]] <math>n\#</math> is the product of [[Prime number|prime number]]s less than or equal {{nowrap|to <math>n</math>;}} this construction gives them some similar divisibility properties to factorials,<ref name=caldwell-gallot/> but unlike factorials they are squarefree.<ref>{{cite book | last = Nelson | first = Randolph | doi = 10.1007/978-3-030-37861-5 | isbn = 978-3-030-37861-5 | location = Cham | mr = 4297795 | page = 127 | publisher = Springer | title = A Brief Journey in Discrete Mathematics | url = https://books.google.com/books?id=m8PPDwAAQBAJ&pg=PA127 | year = 2020| s2cid = 213895324 }}</ref> As with the [[Factorial prime|factorial prime]]s {{nowrap|<math>n!\pm 1</math>,}} researchers have studied [[Primorial prime|primorial prime]]s {{nowrap|<math>n\#\pm 1</math>.<ref name=caldwell-gallot/>}}
;Subfactorial
:The subfactorial yields the number of [[Derangement|derangement]]s of a set of <math>n</math> objects. It is sometimes denoted <math>!n</math>, and equals the closest integer {{nowrap|to <math>n!/e</math>.{{sfn|Graham|Knuth|Patashnik|1988|p=195}}}}
;Superfactorial
:The [[Superfactorial|superfactorial]] of <math>n</math> is the product of the first <math>n</math> factorials. The superfactorials are continuously interpolated by the [[Barnes G-function]].<ref>{{cite journal|last=Barnes|first=E. W.|jfm=30.0389.02|journal=The Quarterly Journal of Pure and Applied Mathematics|pages=264–314|title=The theory of the {{mvar|G}}-function|url=https://gdz.sub.uni-goettingen.de/id/PPN600494829_0031?tify={%22pages%22:[268],%22view%22:%22toc%22}|volume=31|year=1900}}</ref>

==References==
{{Reflist}}

==External links==
* {{OEIS el|A000142|Factorial numbers}}
* {{springer|title=Factorial|id=p/f038080|mode=cs1}}
* {{MathWorld|urlname=Factorial|title=Factorial}}

{{Calculus topics}}
{{Series (mathematics)}}

[[Category:Combinatorics]]
[[Category:Gamma and related functions]]
[[Category:Factorial and binomial topics]]
[[Category:Unary operations]]

{{Sourceattribution|Factorial}}