**Introduction:**

In statistics **factorial** of positive number n, it is denoted n!. It is a product of all non negative numbers which is less than or equal to n. For example, factorial value for 4
is,

**4!= 1×2×3×4=24.**

By default 0! is defined as the value 1.

Factorial process is used in many different places like in combinatorics, algebra and mathematical analysis. Its most essential incidence is fact that there are n! ways to place n distinct objects into a series.

The definition of the statistics factorial can be extended to non-integer arguments, as maintain its most significant properties.

The statistics factorial function expressed as,

It also recursively expressed as,

`n! ={(1 if n=0),((n-1)!*n if n>0):}`

Above two formulas integrate the command,

**0!=1,**

in the first case by the convention that the product of no numbers at all is 1. This is because,

- Accurately one permutation of zero objects the recurrence relation.
- The recurrence relation (n + 1)! = n! × (n + 1), applicable for n > 0, expand to n = 0;
- it permit for the expression of more formulas, such as the exponential function as a power series:

`e^x=sum_(n=0)^oo(x^n)/(n!)`

- it construct many characteristics in combinatorics suitable for all related sizes. There are many ways select 0 from the null set is

`((0),(0))=(0!)/(0!0!)=1`

- There are n! number of ways of assemble n different objects into a series, the permutations of those objects.

- By various reasons the factorial present in algebra, like by means of already declare coefficients of the binomial formula, or by averaging over permutations for symmetrization of particular process.
- Factorials is also used in calculus.

- Factorials are also used broadly in probability theory.

- Factorials used for many purpose in number theory. In, n! is essentially divisible by every prime numbers up to and as well as n. As an effect, n > 5 is a composite number if and only if
**(n-1)!≡0 (mod n).**

- By Wilson’s theorem, the result is strongly defined as,

** (p-1)!≡-1 (mod
p)** if and only if p- prime number.