Factorialof an integer n (written as n! and read as "n factorial") is simply a product of the number "n" and all natural numbers smaller than n. That is,

n! = n × (n-1) × (n-2) × … × 3 × 2 × 1

Note :Remember that factorial of 0 is 1. (0! = 1)

5! = 5 × 4 × 3 × 2 × 1

5!= 120

**Note :** 5! can also be written as 5 × 4!

Therefore, n! = n × (n-1)!

n |
n! |
n (n-1)! |
---|---|---|

1 | 1 | = 1 x 0! |

2 | 2 x 1 | = 2 x 1! |

3 | 3 x 2 x 1 | = 3 x 2! |

4 | 4 x 3 x 2 x 1 | = 4 x 3! |

5 | 5 x 4 x 3 x 2 x 1 | = 5 x 4! |

6 | 6 x 5 x 4 x 3 x 2 x 1 | = 6 x 5! |

Using the rule: n! = n × (n-1)!

11! = 11 × 10!

11! =11 × 3,628,800

11! = 39,916,800

Factorials are most commonly used in "Permutations and Combinations", a branch of Mathematics usually studied under "Probability". However, other areas of mathematics use factorials as well.

Permutations and Combinations are represented by following expression :

` (m!)/(n!)`

Where both m, n are integers.

`(12!)/(8!) ` = `(12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)/(8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) ` …(1)

After cancelling like terms in the numerator and the denominator:

`(12!)/(8!)` = `(12 × 11 × 10 × 9)` = `11,880 `

Note from equation (1) that:

`(12!)/(8!)` = `(12×11×10×9×8!)/(8!)`

Factorials grow exponentially. It is said that in some cases, the growth of factorials is faster than that of exponentials.

n |
n! |
---|---|

1 | 1 |

2 | 1 |

2 | 2 |

3 | 6 |

4 | 24 |

5 | 120 |

6 | 720 |

7 | 5,040 |

8 | 40,320 |

9 | 362,880 |

10 | 3,628,800 |

11 | 39,916,800 |

12 | 479,001,600 |

13 | 6,227,020,800 |

14 | 87,178,291,200 |

15 | 1,307,674,368,000 |

16 | 20,922,789,888,000 |

17 | 355,687,428,096,000 |

18 | 6,402,373,705,728,000 |

19 | 121,645,100,408,832,000 |

20 | 2,432,902,008,176,640,000 |

21 | 51,090,942,171,709,440,000 |

22 | 1,124,000,727,777,607,680,000 |

23 | 25,852,016,738,884,976,640,000 |

24 | 620,448,401,733,239,439,360,000 |

25 | 15,511,210,043,330,985,984,000,000 |

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