Factorial

The factorial of an integer is the product of all integers that are less than or equal to the integer. Factorials are denoted with the symbol "!" written after the integer. For example, "five factorial" would be written as "5!" and evaluated as follows:

5! = 5 × 4 × 3 × 2 × 1 = 120

The factorial of 0 is 1:

0! = 1

The factorial of 1 is also 1, as it is technically 1! = 1 × 0! = 1 × 1 = 1.

Simplifying factorials

Operations involving factorials follow basic arithmetic and algebra rules. The difficulty with factorials is that their values increase very rapidly, so we end up dealing with very large numbers that are difficult to deal with by hand, but the process is not really any different from what we're accustomed to with algebra and arithmetic.

Dividing factorials is also not particularly different, but it may not be immediately obvious. To simplify factorials that are being divided, first identify the larger factorial, then break it up into a product of factorials that enables us to cancel out the smaller factorials involved. This is more or less the same as factoring and simplifying algebraic expressions, or simplifying fractions.

Examples

1. Simplify :

2. Simplify :

Applications of factorials

Factorials are used in many different areas of mathematics, but have their roots in combinatorics. Combinatorics is an area of mathematics that, very generally, deals with counting. Basic examples include combinations and permutations.

Algebra

In algebra, factorials are used as part of the bimomial theorem, which is used for expanding binomials.

The is referred to as the binomial coefficient, and is a type of combination that can be evaluated using the following formula involving factorials:

Other areas of mathematics

Beyond algebra and probability, factorials are used as part of Taylor series in calculus, number theory, and more, but this site will primarily focus on their use within the context of probability, permutations, and combinations.