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

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combinatorics Figure 1: Ferrers' partitioning diagram for 14.

In combinatorics, the formula $\frac{n!}{r! (n - r)!}$ counts the number of ways to select a subset of $r$ objects from a set of $n$ objects where the order of selection does not matter.12

To understand how this formula works, it is helpful to first consider permutations, where order is significant. The number of ways to select and arrange $r$ objects from $n$ is given by the permutation formula:
$_{n} P_{r} = \frac{n!}{(n - r)!}$

Because a combination ignores order, the permutation formula overcounts the number of unique subsets. Specifically, any single group of $r$ objects can be arranged in $r!$ different ways. Since all $r!$ of these arrangements represent the same combination, we must divide the total number of permutations by $r!$ to eliminate these redundant orderings.12

This results in the combination formula, often denoted as $(\binom{n}{r})$ or $n C r$ :
$_{n} C_{r} = \frac{n!}{r! (n - r)!}$

These values are also known as binomial coefficients, as they appear in the expansion of binomial expressions.12