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How can binomial coefficients be found using factorials?

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combinatorics Figure 1: Ferrers' partitioning diagram for 14.

In combinatorics, binomial coefficients—often denoted as $(\binom{n}{k})$ or $_{n} C_{k}$ —represent the number of ways to select a subset of $k$ objects from a set of $n$ objects without regard to order.12

These coefficients can be calculated using factorials with the following formula:

$(\binom{n}{k}) = \frac{n!}{k! (n - k)!}$

In this expression, $n!$ (n factorial) is the product of all positive integers from 1 up to $n$ , and $0!$ is defined as 1.12

The formula is derived from the concept of permutations, where the number of ways to select $k$ objects from $n$ when order matters is given by $\frac{n!}{(n - k)!}$ . Because combinations do not consider the order of selection, and there are $k!$ ways to arrange any set of $k$ objects, the permutation formula is divided by $k!$ to account for these indistinguishable arrangements.2