Factorial

The factorial is a function applied to whole numbers, defined as $n!=\prod _{i=1}^{n}i=n\cdot (n-1)\cdot \ldots \cdot 4\cdot 3\cdot 2$. Before the notation $n!$ was invented, $n$ was a common notation. Its most common usage is in determining the number of permutations n distinct objects can be arranged, because there are n ways to arrange the first object, $n-1$ ways to arrange the second object (as the first spot has been taken), and so forth.

For example,

$5!=5\cdot 4\cdot 3\cdot 2\cdot 1=120$.

The special case $0! = 1$ has been set by definition as there is one way to arrange zero objects.

It can be defined recursively as $0! = 1$ and $n!=n\cdot (n-1)!$.

The first few values of $n!$ for $n=0$, 1, 2, 3, 4, ... are 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, and 39916800.

The sum of the reciprocals of the factorials, $\sum _{i=0}^{\infty }i!={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\ldots$, approaches e, the mathematical constant.

Because $n!=\Gamma(n+1)$ (where $\Gamma(x)$ is the gamma function), $n!=\int _{0}^{\infty }e^{-t}\cdot t^{n+1}dt$. This identity gives us factorials of positive real numbers, not limited to integers:

• $\left({\frac {1}{2}}\right)!={\frac {\sqrt {\pi }}{2}}$
• $\left(-{\frac {1}{2}}\right)!={\sqrt {\pi }}$

Variation

Aalbert Torsius defines a variation on the factorial, where $x!n = \prod^{x}_{i = 1} i!(n - 1) = 1!(n - 1) \cdot 2!(n - 1) \cdot \ldots \cdot x!(n - 1)$ and $x!0 = x$.

$x!n$ is pronounced "nth level factorial of x."

The special case $x!x$ is a function known as the Torian.

See also

• double factorial
• exponential factorial
• falling factorial
• multifactorial
• q-Factorial
• rising factorial
• subfactorial, also known as the left factorial
• Torian