Editing Factorial (section)

{{Short description|Product of numbers from 1 to n}}
{{Use mdy dates|cs1-dates=ly|date=December 2021}}
{{about|a mathematical function|the game|Factorio|other uses}}
{{Good article}}
{| 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]], the '''factorial''' of a non-negative {{nowrap|[[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]].<ref name="gkp">{{cite book|first1=Ronald L.|last1=Graham|author1-link=Ronald Graham |first2=Donald E.|last2=Knuth|author2-link=Donald Knuth|first3=Oren|last3=Patashnik|author3-link=Oren 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 [[Jain literature]], and by Jewish mystics in the Talmudic book ''[[Sefer Yetzirah]]''. The factorial operation is encountered in many areas of mathematics, notably in [[combinatorics]], where its most basic use counts the possible distinct [[sequence]]s – the [[permutation]]s – of <math>n</math> distinct objects: there {{nowrap|are <math>n!</math>.}} In [[mathematical analysis]], factorials are used in [[power series]] for the [[exponential function]] and other functions, and they also have applications in [[algebra]], [[number theory]], [[probability theory]], and [[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]]. [[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. [[Daniel Bernoulli]] and [[Leonhard Euler]] [[interpolate]]d the factorial function to a continuous function of [[complex number]]s, except at the negative integers, the (offset) [[gamma function]].

Many other notable functions and number sequences are closely related to the factorials, including the [[binomial coefficient]]s, [[double factorial]]s, [[falling factorial]]s, [[primorial]]s, and [[subfactorial]]s. Implementations of the factorial function are commonly used as an example of different [[computer programming]] styles, and are included in [[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]]s for numbers with the same number of digits.