Factorial

The factorial is one of the simplest yet most powerful tools in counting. It compactly represents the number of ways to arrange a set of distinct objects in a line — the very foundation of permutations. For any natural number $n$, the notation $n!$ (read as "n factorial") means the product of all positive integers from $1$ up to $n$: $n! = 1 \times 2 \times 3 \times \dots \times n$. By convention, $0!$ is defined …