Recent Articles

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!.

Definition

The factorial function is formally defined by

<math>n!=\prod_{k=1}^n k\qquad\mbox{for all }n\ge0.<math>

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 above recursive relation 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 values of z), via the gamma function:

<math>z!=\Gamma(z+1)=\int_{0}^{\infty} t^z e^{-t}\, dt<math>

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

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800,...

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 permutations.) 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

<math>{n\choose k}={n!\over k!(n-k)!}.<math>

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)!

Calculating factorials

The numeric value of n! can be calculated by repeated multiplication if n is not too large. That is basically what pocket calculators do. The largest factorial that most calculators can handle is 69!, because 70! > 10¹⁰⁰.

When n is large, n! can be estimated quite accurately using Stirling's approximation:

<math>n!\approx\sqrt{2\pi n}\left(\frac{n}{e}\right)^n<math>

Generalizations

The gamma function

The related gamma function Γ(z) is defined for all complex numbers z except for the nonpositive integers (z = 0, −1, −2, −3, ...). It is related to factorials in that it satisfies a recursive relationship similar to that of the factorial function:

<math>n!=n(n-1)!<math>

<math>\Gamma(n+1)=n\Gamma(n)<math>

Together with the definition Γ(1) = 1 this yields the equation

<math>\Gamma(n+1)=n!\qquad\mbox{for all }n\in\mathbb{N},n\ge1.<math>

Because of this relationship, the gamma function is often thought of as a generalization of the factorial function to the domain of complex numbers. This is justified for the following reasons.

• Shared meaning—The canonical definition of the factorial function is the mentioned recursive relationship, shared by both.
• Uniqueness—The gamma function is the only function which satisfies the mentioned recursive relationship for the domain of complex numbers and is holomorphic and whose restriction to the positive real axis is 8, 15, 48, 105, 384, 945, 3840, ...

Some identities involving double factorials are:

<math>n!=n!!(n-1)!!<math>

<math>(2n)!!=2^nn!<math>

<math>(2n+1)!!={(2n+1)!\over(2n)!!}={(2n+1)!\over2^nn!}<math>

<math>\Gamma\left(n+{1\over2}\right)=\sqrt\pi{(2n-1)!!\over2^n}<math>

One should be careful not to interpret n!! as the factorial of n!, which would be written (n!)! and is a much larger number.

The double factorial is the most commonly used variant, but one can similarly define the triple factorial (n!!!) and so on. In general, the k-th factorial, denoted by n!^(k), is defined recursively as

<math>