Factorial !
Example: 4! is shorthand for 4 × 3 × 2 × 1
The factorial function (symbol: !) says to multiply all whole numbers from our chosen number down to 1. Examples:

Calculating From the Previous Value
We can easily calculate a factorial from the previous one:
As a table:
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 
 To work out 6!, multiply 120 by 6 to get 720
 To work out 7!, multiply 720 by 7 to get 5040
 And so on
Example: 9! equals 362,880. Try to calculate 10!
10! = 10 × 9!
10! = 10 × 362,880 = 3,628,800
So the rule is:
n! = n × (n−1)!
Which says
"the factorial of any number is that number times the factorial of (that number minus 1)"
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 multiplying no numbers together results in 1, but let's follow the pattern backwards from, say, 4! like this:
And in many equations using 0! = 1 just makes sense.
Example: how many ways can we arrange letters (without repeating)?
 For 1 letter "a" there is only 1 way: a
 For 2 letters "ab" there are 1×2=2 ways: ab, ba
 For 3 letters "abc" there are 1×2×3=6 ways: abc acb cab bac bca cba
 For 4 letters "abcd" there are 1×2×3×4=24 ways: (try it yourself!)
 etc
The formula is simply n!
Now ... how many ways can we arrange no letters? Just one way, an empty space:
So 0! = 1
Where is Factorial Used?
One area they are used is in Combinations and Permutations. We had an example above, and here is a slightly different example:
Example: How many different ways can 7 people come 1^{st}, 2^{nd} and 3^{rd}?
The list is quite long, if the 7 people are called a,b,c,d,e,f and g then the list includes:
abc, abd, abe, abf, abg, acb, acd, ace, acf, ... etc.
The formula is \frac{7!}{(7−3)!} = \frac{7!}{4!}
Let us write the multiplies out in full:
\frac{7 × 6 × 5 × 4 × 3 × 2 × 1}{4 × 3 × 2 × 1} = 7 × 6 × 5
That was neat. The 4 × 3 × 2 × 1 "cancelled out", leaving only 7 × 6 × 5. And:
7 × 6 × 5 = 210
So there are 210 different ways that 7 people could come 1^{st}, 2^{nd} and 3^{rd}.
Done!
Example: What is 100! / 98!
Using our knowledge from the previous example we can jump straight to this:
\frac{100!}{98!} = 100 × 99 = 9900
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 the Full Precision Calculator.
Interesting Facts
Six weeks is exactly 10! seconds (=3,628,800)
Here is why:
Seconds in 6 weeks:  60 × 60 × 24 × 7 × 6  
Factor some numbers:  (2 × 3 × 10) × (3 × 4 × 5) × (8 × 3) × 7 × 6  
Rearrange:  2 × 3 × 4 × 5 × 6 × 7 × 8 × 3 × 3 × 10  
Lastly 3×3=9:  2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 
There are 52! ways to shuffle a deck of cards.
That is 8.0658175... × 10^{67}
Just shuffle a deck of cards and it is likely that you are the first person ever with that particular order.
There are about 60! atoms in the observable Universe.
60! is about 8.320987... × 10^{81} and the current estimates are between 10^{78} to 10^{82} atoms in the observable Universe.
70! is approximately 1.197857... 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 Topics
A Close Formula!
The "≈" means "approximately equal to". Let us see how good it is:
n  n!  Close Formula (to 2 Decimals)  Accuracy (to 4 Decimals) 

1  1  0.92  0.9221 
2  2  1.92  0.9595 
3  6  5.84  0.9727 
4  24  23.51  0.9794 
5  120  118.02  0.9835 
6  720  710.08  0.9862 
7  5040  4980.40  0.9882 
8  40320  39902.40  0.9896 
9  362880  359536.87  0.9908 
10  3628800  3598695.62  0.9917 
11  39916800  39615625.05  0.9925 
12  479001600  475687486.47  0.9931 
If you don't need perfect accuracy this may be useful.
Note: it is called "Stirling's approximation" and is based on a simplifed version of the Gamma Function.
What About Negatives?
Can we have factorials for negative numbers?
Yes ... but not for negative integers.
Negative integer factorials (like 1!, 2!, etc) are undefined.
Let's start with 3! = 3 × 2 × 1 = 6 and go down:
2!  =  3! / 3  =  6 / 3  =  2  
1!  =  2! / 2  =  2 / 2  =  1  
0!  =  1! / 1  =  1 / 1  =  1  which is why 0!=1  
(−1)!  =  0! / 0  =  1 / 0  =  ?  oops, dividing by zero is undefined 
And from here on down all integer factorials are undefined.
What About Decimals?
Can we have factorials for numbers like 0.5 or −3.217?
Yes we can! But we need to use the Gamma Function (advanced topic).
Factorials can also be negative (except for negative integers).
Half Factorial
But I can tell you the factorial of half (½) is half of the square root of pi .
Here are some "halfinteger" factorials:
(−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?
Double Factorial!!
A double factorial is like a normal factorial but we skip every second number:
 8!! = 8 × 6 × 4 × 2 = 384
 9!! = 9 × 7 × 5 × 3 × 1 = 945
Notice how we multiply all even, or all odd, numbers.
Note: if we want to apply factorial twice we write (n!)!