How does the formula n! divided by (n-r)! count permutations?

In combinatorics, a permutation is an arrangement of objects where the order of selection is significant. The formula $\frac{n!}{(n-r)!}$ counts the number of ways to select and arrange $r$ objects from a set of $n$ distinct objects.1

To understand why this formula works, consider the process of selecting $r$ items one by one:
* For the first position, there are $n$ choices.
* For the second, there are $n-1$ choices.
* This continues until the $r$-th position, which has $n-(r-1)$ or $n-r+1$ choices.

Multiplying these choices together gives $n\times (n-1)\times \dots \times (n-r+1)$. The factorial notation $n!$ represents the product of all integers from $1$ to $n$. By dividing $n!$ by $(n-r)!$, you effectively cancel out the product of all integers from $1$ to $n-r$, leaving exactly the product of the $r$ terms required for the permutation.1

This counting formula allows you to determine the total number of ordered arrangements without the need to list every possibility.1