How does the factorial formula calculate the number of arrangements of n distinct objects?

In combinatorics, the factorial formula, denoted as $n!$, calculates the total number of ways to arrange $n$ distinct objects in a row.

This formula works by multiplying all consecutive positive integers from 1 up to and including $n$:
$$n!=n\times (n-1)\times (n-2)\times \dots \times 1$$

The logic behind this is based on the sequential choices available when placing objects:
* For the first position, there are $n$ possible objects to choose from.
* For the second position, there are $n-1$ objects remaining.
* This continues until only one object remains for the final position.

By the fundamental counting principle, the total number of permutations is the product of these choices, resulting in $n!$. For example, if you have 5 distinct objects, there are $5!=5\times 4\times 3\times 2\times 1=120$ possible arrangements. By convention, $0!$ is defined as 1.12