A menu lists five vegetable; applesauce, beans, carrots, peas, and potatoes. A person wants to order a vegetable plate with a combination of any 3 different vegetables from the menu. The total number of choices that the person has can be found using combinatorics, a branch of mathematics dealing with combinations and permutations.
Given: n (items in menu) = 5, r (combination of 3 vegetables) = 3
Substituting the value for n and r in the expression for calculating the available choices, [tex] ^{n}C_{r} = \frac{n!}{(n-r)!r!} [/tex],
We get,
[tex] ^{5}C_{3} = \frac{5!}{(5-3)!\times3!} = \frac{5\times4\times3!}{2!\times3!} = \frac{5\times4}{2} = 10 [/tex]
Therefore, the answer is 10 choices or unique combinations.
