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Home > Factorial

In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. This is written as n! and pronounced "n factorial". The notation n! was introduced by Christian Kramp in 1808.

Since the exclamation mark, "!", is sometimes pronounced "bang" or "shriek", these words are occasionally used colloquially for "factorial" in pronouncing equations like n!.

1 Definition

The factorial function is formally defined by

For example,

5! = 1 × 2 × 3 × 4 × 5 = 120

This definition implies in particular that

0! = 1

because the product of no numbers at all is 1 (see empty product for an account of that fact). Proper attention to the value of the empty product is important in this case, because

• it makes the recursive relation (n + 1)! = n!(n + 1) work for n = 1;
• many identities in combinatorics would not work for zero sizes without this definition.

The factorial function can also be defined (for non-integer in addition to the usual integer values of z), via the gamma functionIn mathematics, the gamma function is a function that extends the concept of factorial to the complex numbers. Definition The notation Γ z is due to Adrien-Marie Legendre. If the real part of the complex number z is positive, then the integral : con:

The sequence of factorials (sequence A000142 in OEIS) for n = 0, 1, 2,... starts:

1, 1, 2, 66 six is the natural number following 5 and preceding 7. Prefixes for 6 include hexa- ( Greek) and sex- ( Latin). The SI prefix for 10006 is exa (E), and for its reciprocal atto (a). Evolution of the glyph The evolution of our modern glyph for 6 appears r, 242 4 2 4 24 twenty-four is the natural number following 23 and preceding 25. Cardinal twenty-four Ordinaltwenty-fourth Factorization Roman numeralXXIV Binary11000 Hexadecimal18 In mathematics It is the factorial of 4 and a composite number, its proper divi, 120120 one hundred twenty in American English; one hundred and twenty in British English) is the natural number following 119 and preceding 121. 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled, 72079e02 720 Seven hundred twenty is the natural number following 719 and preceding 721. 720 CardinalSeven hundred and twenty Ordinal720th Factorization Roman numeralDCCXX Binary1011010000 Hexadecimal2D0 It is 6! ( 6 factorial), a composite number with lots, 5040, 40320, 362880, 3628800,...

2 Applications

Factorials are important in combinatorics. For example, there are n! different ways of arranging n distinct objects in a sequence. (The arrangements are called permutationThis article is about permutation a mathematical concept. See permutation (music) for the application of this concept to music. In mathematics, the concept of a permutation expresses the idea that objects that can be distinguished may be arranged in varios.) And the number of ways one can choose k objects from among a given set of n objects (the number of combinations), is given by the so-called binomial coefficient

Factorials also turn up in calculus. For example, Taylor's theorem expresses a function f(x) as a power series in x, basically because the n-th derivative of xⁿ is n!. Factorials are also used extensively in probability theory.

Factorials are often used as a simple example when teaching recursion in computer science because they satisfy the following recursive relationship (if n ≥ 1):

n! = n (n − 1)!

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Topics: Factorial Function Factorials Sequence Defined Product Number...

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FactorialNumber theory Combinatorics In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n''. This is written as n and pronounced n factorial". The notation n was introduced by Christian Kramp in 1808. Factorial primeA factorial prime is a number that is one less or one more than a factorial and is also a prime number. The first few factorial primes are: 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199,. Factorial primes are of interest to number theorists Factorial momentIn probability theory, the n''th factorial moment of a probability distribution, also called the n''th factorial moment of any random variable X with that probability distribution, is : where : is the falling factorial (confusingly, this same notation, th