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 × (n1)!
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!
What About "0!"
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 10^{100}, which is just larger than a Googol (the digit 1 followed by one hundred zeros).
100! is approximately 9.3326215443944152681699238856 x 10^{157}
200! is approximately 7.8865786736479050355236321393 x 10^{374}
What About Decimals?
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 "halfinteger" 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?
