Editing Factorial (section)

==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 [[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 | author1-link = Bibhutibhushan Datta | 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| isbn = 978-981-13-7325-1 | 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 [[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 [[Shankha|conch shell]], [[Sudarshana Chakra|discus]], [[Kaumodaki|mace]], and [[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. |author-link=Norman L. Biggs |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 ''[[Sefer Yetzirah]]'', from the [[Talmud|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 [[Hebrew alphabet]].<ref name=katz>{{cite journal | last = Katz | first = Victor J. | author-link = Victor J. Katz | 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 [[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]]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 [[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=[[Oxford University Press]]|date=2013|isbn=978-0-19-965659-2|first=Victor J.|last=Katz|author-link=Victor J. Katz|contribution=Chapter 4: Jewish combinatorics|pages=109–121}} See p. 111.</ref> In 1677, British author [[Fabian Stedman]] described the application of factorials to [[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 [[Ancient Society of College Youths|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 [[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=[[Oxford University Press]]|date=2013|isbn=978-0-19-965659-2|first=Eberhard|last=Knobloch|author-link=Eberhard Knobloch|contribution=Chapter 5: Renaissance combinatorics|pages=123–145}}  See p. 126.</ref> [[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.{{sfn|Knobloch|2013|pages=130–133}} 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]].<ref name=exponential-series>{{cite book | last1 = Ebbinghaus | first1 = H.-D. | author1-link = Heinz-Dieter Ebbinghaus | last2 = Hermes | first2 = H. | author2-link = Hans Hermes | last3 = Hirzebruch | first3 = F. | author3-link = Friedrich Hirzebruch | 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 [[John Wallis]], a study of their approximate values for large values of <math>n</math> by [[Abraham de Moivre]] in 1721, a 1729 letter from [[James Stirling (mathematician)|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]].<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> [[Adrien-Marie Legendre]] included [[Legendre's formula]], describing the exponents in the [[Integer factorization|factorization]] of factorials into [[prime power]]s, in an 1808 text on [[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 | author-link = Florian Cajori | 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]]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>