Calculating Individual Factorials
A factorial, denoted as $n!$, is a mathematical operation that multiplies a given number $n$ by all positive integers less than itself. This concept is fundamental in combinatorics and various fields of mathematics and computer science. The factorial of a nonnegative integer $n$ can be defined recursively or iteratively.
Definition
Formally, the factorial of $n$ is given by:
$n! =
\begin{cases}
1 & \text{if } n = 0 \\
n \times (n1) \times (n2) \times \cdots \times 1 & \text{if } n > 0
\end{cases}$
Examples
 The factorial of 5 ($5!$) is:
$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
 The factorial of 0 ($0!$) is defined as 1 by convention.
Properties

Growth Rate: Factorials grow exponentially. For large $n$, $n!$ becomes very large, very quickly.

Recursive Definition:
$n! = n \times (n1)!$
Usage in Permutations and Combinations
Factorials are crucial in calculating permutations and combinations in probability and statistics. For instance, the number of ways to arrange $n$ distinct objects is $n!$.
Implementation
Recursively:
$\text{factorial}(n) =
\begin{cases}
1 & \text{if } n = 0 \\
n \times \text{factorial}(n1) & \text{if } n > 0
\end{cases}$
Iteratively:
$\text{factorial} = 1 \\
\text{for } i \text{ in range(1, n+1)}: \\
\text{ factorial} *= i$
Understanding and calculating factorials is essential for solving many mathematical problems and performing specific computational tasks efficiently.