The Factorial Function
The factorial function is defined as:
F(n) = n(n  1)(n  2)(n  3)^{ ... }(2)(1)
where n is a nonnegative integer.
We define
F(0) = 1 and
F(1) = 1.
The factorial function F(n) is also represented as "n!", read "n
factorial."
Examples.
5!  =  5(4)(3)(2)(1) = 120 

3!  =  3(2)(1) = 6 

10!  =  10(9)(8)(7)(6)(5)(4)(3)(2)(1) = 3628800 

0!  =  1;;(by definition) 

6!  =  6(5)(4)(3)(2)(1) = 720 

Example 1: If 6 children must form a line, in how many ways can they
arrange themselves?
6 different children can stand in the first spot, 5 in the second spot (since 1
is already occupying the first spot), 4 in the third spot (since 2 are already
occupying spots), etc.
Thus, they can arrange themselves in 6(5)(4)(3)(2)(1) = 6! = 720 different ways.
n! gives the number of ways n distinct items can be arranged in an order.
Example 2: In how many ways can the letters of the word TRIANGLE be
arranged?
8! = 40320 different ways.