Factorial !
Example: 4! is shorthand for 4 x 3 x 2 x 1
The factorial function (symbol: !) means to multiply a series of descending natural numbers. Examples:

Calculating From the Previous Value
We 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: 9!=362,880. Work out the value of 10!
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 that number times the factorial of (1 smaller than that number)", so 10! = 10 × 9!, ... and 125! = 125 × 124!, etc.
What About "0!"
Zero Factorial is interesting ... it is generally agreed that 0! = 1.
It may seem funny that in this case multiplying no numbers together results in 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
Example: What is 7! / 4!
Let us write them out in full:
7 × 6 × 5 × 4 × 3 × 2 × 1 
= 7 × 6 × 5 = 210 
4 × 3 × 2 × 1 
That was neat. The 4 × 3 × 2 × 1 "cancelled out", leaving only 7 × 6 × 5
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  51,090,942,171,709,440,000 
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 
As you can see, it gets big quickly.
If you need more, try our Full Precision Calculator.
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}
Advanced Topic
What About Decimals?
Can we have factorials for numbers like 0.5 or 3.217?
Yes we can! But we 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 
It still follows the rule that "the factorial of any number is that number times the factorial of (1 smaller than that number)", because
(3/2)! = (3/2) × (1/2)!
(5/2)! = (5/2) × (3/2)!
Can you figure out what (7/2)! is?