# Introduction

The factorial of a positive integer which is generally represented or denoted as n! or a particular positive number n can be defined as the product of every positive number equivalent to or preceding n that is the positive integer. Factorial is essentially a function whose use can be seen in several areas of mathematics such as algebra, business mathematics, combinatorics, and mathematical analysis.

## Factorial of a number

Factorial of a positive number or integer can be defined as the product of every positive integer before or/ and equal to the integer n. mathematically it is denoted as

n! = 1. 2. 3…. (n-2). (n-1) . n

If an individual is looking at integers or values equal to or greater than 1, then it can be written as the recurrence relation of n! = n. (n-1)!.

Some examples can be sighted in this context.

- 5! = 5 . 4!
- 9! = 9 . 8!
- 79! = 79. 78!

In the above example, 5 factorial can be calculated as 5! = 5 * 4 *3 *2 *1 = 120. The calculation of the 5 factorial is quite normal.

## Factorial table

Factorial of any number can be easily calculated by using the above formula. However, a table has been provided in the following section showing the factorial value of the integers from 0 to 11.

Positive integer (n) | n! or factorial n |
---|---|

0 | one |

1 | one |

2 | two |

3 | six |

4 | Twenty-four |

5 | One Twenty |

6 | Seven twenty |

7 | Five thousand forty |

8 | Forty thousand three hundred twenty |

9 | Three hundred sixty two thousand eight hundred eighty |

10 | Three million six hundred twenty-eight thousand eight hundred |

11 | Thirty nine million nine hundred sixteen thousand eight hundred |

## 0! or zero factorial

It is a known fact that 0 factorial is given as 1 and it is denoted by 0! = 1. Many reasons can be given to justify this result. Firstly from the meaning of factorial, it is clear that the factorial of a number provides an expression that is compact of a particular number of formulas involving exponential functions, the definition itself creates a particular extension to the zero’s recurrence relation. Moreover when n is taken as 0, the meaning of n! Suggests the product has no number. Hence, this is equal to the multiplicative identity in rather broad terms. Thirdly, zero factorials definition involves particularly one permutation of 0 that has no objects. Finally, the meaning of zero factorial also identifies several identities within combinatorics.

## Computation

The function of factorial is often used in scientific calculators. It is further included within the module of Python mathematical functions as well as the library of Boost C++. If we do not consider efficiency, then the calculation of factorial is vital. To calculate this, a variable that is initialized to 1 is repeatedly multiplied with the successive integers till n. The simple method of calculation of this function makes it a rather common example for using within different computer programming methods and styles. Other methods through which suitable computation of the function can be done involve dynamic programming, functional programming, and memorization.

## Different applications of factorial of a number

The function of factorial can be seen in several fields within mathematics. At first, the ‘n!’ distinct ways can be noticed to arrange n specific items in the form of a sequence. In addition, factorials can also be used for accounting for the disregard or ignorance of ordering within a formula through serving as a denominator. Factorials are also seen to be used in algebra through the binomial theorem. Moreover, factorials used in the denominators of Taylor’s theorem. Lastly, a factorial is seen to be applied in the theory of numbers and probability and they can be further used for enabling expression manipulation.

### Conclusion

The chief subject on which the entire article has been written is business mathematics. Under this subject the topic that has been mostly discussed throughout the article is factorial. Factorial is a basic concept in mathematics and under this main concept, the meaning of the term factorial, as well as the definition of factorial of a number, has been discussed. In addition to this, some other items such as the definition of zero factorial, and the wide range of applications of factorial has also been discussed within this research.