Selected factorials; values in scientific notation are rounded
0
1
1
1
2
2
3
6
4
24
5
120
6
720
7
5040
8
40320
9
362880
10
3628800
11
39916800
12
479001600
13
6227020800
14
87178291200
15
1307674368000
16
20922789888000
17
355687428096000
18
6402373705728000
19
121645100408832000
20
2432902008176640000
25
1.551121004×1025
50
3.041409320×1064
70
1.197857167×10100
100
9.332621544×10157
450
1.733368733×101000
1000
4.023872601×102567
3249
6.412337688×1010000
10000
2.846259681×1035659
25206
1.205703438×10100000
100000
2.824229408×10456573
205023
2.503898932×101000004
1000000
8.263931688×105565708
10100
1010101.9981097754820
In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial:
For example,
The value of 0! is 1, according to the convention for an empty product.[1]
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 sequences – the permutations – of distinct objects: there are . 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 interpolated the factorial function to a continuous function of complex numbers, 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 coefficients, double factorials, falling factorials, primorials, and subfactorials. Implementations of the factorial function are commonly used as an example of different computer programming styles, and are included in 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 algorithms for numbers with the same number of digits.
^Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1988). Concrete Mathematics. Reading, MA: Addison-Wesley. p. 111. ISBN 0-201-14236-8.
In mathematics, the factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers...
In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd...
In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels"...
gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex...
mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial...
In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the factorial function developed by the Fields...
fractional factorial designs are experimental designs consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design...
accepted by the emerging field of psychology which developed strong (full factorial) experimental methods to which randomization and blinding were soon added...
A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes...
The exponential factorial is a positive integer n raised to the power of n − 1, which in turn is raised to the power of n − 2, and so on in a right-grouping...
mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement...
the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base...
Number of trailing zeros for any factorial Python program to calculate the number of trailing zeros for any factorial Archived 2017-02-22 at the Wayback...
acceptable mathematically. But different factorial theories proved to differ as much in terms of the orientations of factorial axes for a given solution as in...
the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments...
the factorial of n: function factorial (n is a non-negative integer) if n is 0 then return 1 [by the convention that 0! = 1] else return factorial(n –...
In mathematics, the Fibonorial n!F, also called the Fibonacci factorial, where n is a nonnegative integer, is defined as the product of the first n positive...
In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials of positive integers. This is the same...
the factorial function for non-negative integers, shown in Haskell: factorial :: Integer -> Integer factorial 0 = 1 factorial n = n * factorial (n-1)...
in the data vector. When this occurs, it can be desirable to create a factorial code of the data, i.e., a new vector-valued representation of each data...
the loop factorial = factorial * counter counter = counter - 1 end do print *, factorial end program FactorialProg int counter = 5; int factorial = 1; do...
equation models, including CFA, measurement invariance is often termed factorial invariance. In the common factor model, measurement invariance may be...
subfactorial !n equals the nearest integer to n!/e, where n! denotes the factorial of n and e is Euler's number. The problem of counting derangements was...
Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and superscripts (e.g...
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