Definition of a factorial

The factorial of a non-negative integer number n, denoted as n! is the product of all the integers ranging from 1 to n. That is:

n! = n.(n-1).(n-2)...3.2.1

Examples:

5! = 5.4.3.2.1 = 120

9! = 9.8.7.6.5.4.3.2.1 = 362880

Note that

9! = 9.8.7.6.5!

It means we can shorten the expression of a factorial as much as desired for simplifying purposes

A special rule of factorials state that 0! = 1

Simplifying factorial expressions

The above mentioned properties of the factorials can be used to simplify expressions where several factorials can be found. Let's simplify the expression:

To better simplify the expression, the largest factorial should be expanded until it matches with the largest factorial in the opposite side of the fraction. In the example provided, the 9! will be expanded up to 7! to simplify them both. The remaining 4! must be expanded completely

When the argument of a factorial is a sum, a subtraction, or any other operation, they must be performed before the factorial, since the factorial of a sum is in general, different from the sum of factorials. The same applies to the subtraction and other operations.

Example: (1+3)! = 4! = 4.3.2.1 = 24

(5-5)! = 0! = 1