Factorial

The factorial is a function that is defined only for non-negative integers $n$ . The factorial of a number is written as $n!$ , and is the product of all positive integers less than or equal to $n$ . Important to note is that $0! = 1$ and $1! = 1$ . For $n > 1$ , the factorial is defined as:

n! = n \cdot (n-1) \cdot (n-2)\cdot ... \cdot 2 \cdot 1

or alternatively,

n! = \prod_{x=1}^{n}{x}

Example

$n$	$n!$
$0$	$1$
$1$	$1$
$2$	$2 \cdot 1 = 2$
$3$	$3 \cdot 2 \cdot 1 = 6$
$4$	$4 \cdot 3 \cdot 2 \cdot 1 = 24$
$5$	$5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$
$n$	$n \cdot (n-1) \cdot (n-2)\cdot ... \cdot 2 \cdot 1$

Why is $0! = 1$?

There are many reasons why $0!$ must be $1$ , most of them being related to combinatorics and counting problems. However, there is an intuitive explanation that can be given. Notice above that $n! = n \cdot (n-1)!$ . This means that the factorial of a number is the product of that number and the factorial of the number that is one less than it.

For the above to hold true for $1!$ we need to have $0! = 1$ because if $0!=0$ , then $1!$ would be:

1! = 1 \cdot (1-1)! = 1 \cdot 0! = 1 \cdot 0 = 0

and all other factorials would also be $0$ as well. Therefore, $0!$ must be $1$ .

Sterlings Approximation

n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n

Absolute Value Binomial Coefficient