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←Membuat halaman berisi '{{Short description|Product of numbers from 1 to n}} {{Use mdy dates|cs1-dates=ly|date=December 2021}} {{about|hasil kali bilangan bulat berurutan|statistical experiments over all combinations of values|factorial experiment|data representation by independent components|factorial code}} {| class="wikitable" style="margin:0 0 0 1em; text-align:right; float:right;" |+Selected factorials; nilai dalam notasi ilmiah dibulatkan !<math>n</math> !<math>n!</math> |- |0...'
 
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\end{align}</math>Contohnya,<math display="block">5! = 5 \times 4 \times 3 \times 2 \times 1 = 5\times 24 = 120. </math>Nilai <math>0!</math> adalah 1, menurut konvensi [[darab kosong]].<ref name="gkp">{{cite book|last1=Graham|first1=Ronald L.|last2=Knuth|first2=Donald E.|last3=Patashnik|first3=Oren|date=1988|title=Concrete Mathematics|title-link=Concrete Mathematics|location=Reading, MA|publisher=Addison-Wesley|isbn=0-201-14236-8|page=111|author1-link=Ronald Graham|author2-link=Donald Knuth|author3-link=Oren Patashnik}}</ref>
 
FactorialsFaktorial haveditemukan beendalam discoveredbeberapa inbudaya several ancient cultureskuno, notablykhususnya indi [[Indianmatematika mathematicsIndia]] indalam thetulisan canonical works ofkarya [[sastra Jain literature]], anddan byoleh <u>Jewish mystics</u> dalam inbuku theTalmud Talmudicyang bookberjudul ''[[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|sequences]] – the [[Permutation|permutations]] – 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|interpolated]] the factorial function to a continuous function of [[Complex number|complex numbers]], except at the negative integers, the (offset) [[gamma function]].
 
ManyBanyak otherfungsi notablekhusus functionsdan andbarisan numberbilangan sequenceslainnya areterkait closelyerat relateddengan to the factorialsfaktorial, includingdi theantaranya [[Binomialkoefisien coefficient|binomial coefficients]], [[Doublefaktorial factorial|double factorialsganda]], <u>[[Falling factorial|fallingfaktorial factorialsturun]]</u>, [[Primorial|primorialsprimorial]], anddan [[Subfactorial|subfactorialssubfaktorial]]. Implementations of the factorial function are commonly used as an example of different [[computer programming]] styles, and are included in [[Scientific calculator|scientific calculators]] 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 algorithms]] for numbers with the same number of digits.
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