## Factorial !

The factorial function (using symbol: !) says to multiply all whole numbers from our chosen number down to 1.

So *n*! (read “*n* factorial”) means **n**** × (****n****-1) × (****n****-2) × … × 3 × 2 × 1**

Table of Contents

## Definition of Factorial

What is the Factorial function?

The factorial function is a function represented by an exclamation mark “!”. To find the result of Factorial function, you need to multiply the number that appears in the formula with all the positives integers that smaller than it.

## The utility of the Factorial function

Usually, Factorial functions is used to calculate permutations & combinations, so you can also calculate probabilities.

If you want to hang 5 Photos on the wall, one after another you can calculate the number of possible combinations:

5! = 5×4×3×2×1 = 120 possible combinations

**Other Examples:**

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362880

1! = 1

## Calculating Factorial From the Previous Value

We can easily calculate a factorial from the previous one:

n | n! | n × (n-1)! | Result |

1 | 1 | 1 | 1 |

2 | 2 × 1 | 2 × 1! | 2 |

3 | 3 × 2 × 1 | 3 × 2! | 6 |

4 | 4 × 3 × 2 × 1 | 4 × 3! | 24 |

5 | 5 × 4 × 3 × 2 × 1 | 5 × 4! | 120 |

… | … | … | … |

So the rule is:

**n****! = ****n**** × (****n****−1)!**

Which says

“

the factorial of any number is that number times the factorial of(that number minus 1)”

7! = 7 × 6!

100! = 100 × 99!

etc.

## What About 0! ?

Zero Factorial is interesting

It is generally agreed that 0! = 1.

It may seem funny that multiplying no numbers together results in 1, but let’s start from the rule:

*n*! = *n* × (*n*−1)!

If n = 1, then

1! = 1 × (1−1)!

1 = 1 × 0!

0! = 1/1

0! = 1

So, in many equations using 0! = 1 just makes sense.

## Table of factorial numbers

n | n! | Result |

0 | 1 | 1 |

1 | 1 | 1 |

2 | 2 × 1! | 2 |

3 | 3 × 2! | 6 |

4 | 4 × 3! | 24 |

5 | 5 × 4! | 120 |

6 | 6 × 5! | 720 |

7 | 7 × 6! | 5,04 |

8 | 8 × 7! | 40,32 |

9 | 9 × 8! | 362,88 |

10 | 10 × 9! | 3,628,800 |

11 | 11 × 10! | 39,916,800 |

12 | 12 × 11! | 479,001,600 |

13 | 13 × 12! | 6,227,020,800 |

14 | 14 × 13! | 87,178,291,200 |

15 | 15 × 14! | 1,307,674,368,000 |

16 | 16 × 15! | 20,922,789,888,000 |

17 | 17 × 16! | 355,687,428,096,000 |

18 | 18 × 17! | 6,402,373,705,728,000 |

19 | 19 × 18! | 121,645,100,408,832,000 |

20 | 20 × 19! | 2,432,902,008,176,640,000 |

21 | 21 × 20! | 51,090,942,171,709,440,000 |

22 | 22 × 21! | 1,124,000,727,777,607,680,000 |

23 | 23 × 22! | 25,852,016,738,884,976,640,000 |

24 | 24 × 23! | 620,448,401,733,239,439,360,000 |

25 | 25 × 24! | 15,511,210,043,330,985,984,000,000 |

## History of Factorial

Factorial was used by Indian scholars to calculate the permutations at least as early as the 12^{th} century.

Then, Fabian Stedman described factorials as applied to change ringing, a musical art involving the ringing of many tuned bells In 1677. Stedman gives a statement of a factorial:

“ Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers … in so much that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body”

In 1808, French mathematician Christian Kramp introduced The notation n!.