Editing Factorial (section)

===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]], 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 | author-link = Joel David Hamkins | 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]] 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]] 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 (recursion)|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]], as a [[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]] {{nowrap|<math>0! = \Gamma(0+1) = 1</math>,}} and the gamma function must have this value to be a [[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>