Learn Factorial

Factorial of 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)

Example
 120

Explanation:

5! = 5 × 4 × 3 × 2 × 1
5!= 120

Note : 5! can also be written as 5 × 4!
Therefore, n! = n × (n-1)!


Factorial of basic numbers

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!

Example:
 39,916,800

Explanation:

Using the rule: n! = n × (n-1)!

11! = 11 × 10!

11! =11 × 3,628,800

11! = 39,916,800


Use of Factorials


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.


 11,880

Explanation:

`(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!)`


How Factorials Grow


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


The following table illustrates how fast factorials grow

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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