$\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$ $\text{where } n = \text{total objects},\quad k = \text{objects selected}$ A Combination $\binom{n}{k}$ (read "$n$ choose $kquot;) counts the number of ways to select $k$ objects from $n$ distinct objects when **order does not matter** and **replacement is not allowed**. Combinations appear in the binomial and hypergeometric distributions, in inclusion-exclusion counting, and wherever the identity of the selected group matters but not the order of selection. The identity $\binom{n}{k} = \binom{n}{n-k}$ reflects that choosing $k$ items to include is equivalent to choosing $n-k$ to exclude. > [!example]- Selecting a Claims Committee {💡 Example} > An insurer needs to form a committee of 3 adjusters from a pool of 8. How many different committees are possible? > > > [!answer]- Answer > > Order does not matter (a committee $\{A, B, C\}$ is the same regardless of selection order), so: > > $\binom{8}{3} = \frac{8!}{3!\,5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56$ > > There are 56 possible committees.