Combination Formula

Property The number of combinations of $n$ items taken $r$ at a time is $$ nC r = \frac{n!}{r!(n r)!} $$ Explanation This awesome formula lets you count combinations without listing every single one! It cleverly starts with all the permutations (ordered groups) and then divides by the number of ways to arrange the chosen items ($r!$). This removes all the duplicates, since with combinations, order doesn't matter. It’s a powerful shortcut to find just the unique groups. Examples To find the combinations of choosing 2 side dishes from 6: $ 6C 2 = \frac{6!}{2!(6 2)!} = \frac{6!}{2!4!} = 15$. To find the combinations of choosing 3 test questions from 4: $ 4C 3 = \frac{4!}{3!(4 3)!} = \frac{4!}{3!1!} = 4$. How many ways can a customer choose 12 fruit types from 16 options? $ {16}C {12} = \frac{16!}{12!4!} = 1820$.