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# Factorial !

 The factorial function (symbol: !) just means to multiply a series of descending natural numbers. Examples: 4! = 4 × 3 × 2 × 1 = 24 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 1! = 1

 "4!" is usually pronounced "4 factorial". Some people even say "4 shriek" or "4 bang"

## Calculating From the Previous Value

You can easily calculate a factorial from the previous one:

n n!
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
6 etc etc

### Example: What is 10! if you know that 9!=362,880 ?

10! = 10 × 9!

10! = 10 × 362,880 = 3,628,800

So the rule is:

n! = n × (n-1)!

Which just says "the factorial of any number is: the number times the factorial of (1 smaller than the number)", hence 10! = 10 × 9!, or even 125! = 125 × 124!

Zero Factorial is interesting ... it is generally agreed that 0! = 1.

It may seem funny that multiplying no numbers together gets you 1, but it helps simplify a lot of equations.

## Where is Factorial Used?

Factorials are used in many areas of mathematics, but particularly in Combinations and Permutations

## A Small List

n n!
0 1
1 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 51,090,942,171,709,400,000
22 1,124,000,727,777,610,000,000
23 25,852,016,738,885,000,000,000
24 620,448,401,733,239,000,000,000
25 15,511,210,043,331,000,000,000,000

As you can see, it gets big quickly!

## Some Bigger Values

70! is approximately 1.1978571669969891796072783721 x 10100, which is just larger than a Googol (the digit 1 followed by one hundred zeros).

100! is approximately 9.3326215443944152681699238856 x 10157

200! is approximately 7.8865786736479050355236321393 x 10374

Can you have factorials for numbers like 0.5 or -3.217?

Yes you can! But you need to get into a subject called the "Gamma Function", which is beyond this simple page.

### Half Factorial

But I can tell you the factorial of half (½) is half of the square root of pi = (½)√π, and so some "half-integer" factorials are:

n n!
(-½)! √π
(½)! (½)√π
(3/2)! (3/4)√π
(5/2)! (15/8)√π

And it still follows the rule that "the factorial of any number is: the number times the factorial of (1 smaller than the number)", because

(3/2)! = (3/2) × (1/2)!
(5/2)! = (5/2) × (3/2)!

Can you figure out what (7/2)! is?