Combination

Study guide: Exam P-1

A Combination \binom{n}{k} ("n choose k") counts the number of ways to select k objects from n distinct objects when order does not matter.

• Combinations assume no replacement: you cannot select the same object twice:

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

• n is the total number of objects and k is the number of objects selected
• The identity \binom{n}{k} = \binom{n}{n-k} reflects that choosing k items to include is equivalent to choosing n-k to exclude
• Combinations appear in the Binomial distribution, Hypergeometric distribution, and the Inclusion-Exclusion Principle for counting

Selecting a Claims Committee An insurer needs to form a committee of 3 adjusters from a pool of 8. How many different committees are possible?

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.