Question

How many ways are there to choose a dozen apples from a bushel containing\({\rm{20}}\)indistinguishable delicious apples,\({\rm{20}}\)indistinguishable Macintosh apples, and\({\rm{20}}\)indistinguishable Granny Smith apples, if at least three of each kind must be chosen?

Short answer

There are \({\rm{10}}\)ways to choose a dozen apples.

Step-by-step solution

Introduction

A combination in mathematics is a selection of elements from a set with distinct members, where the order of selection is irrelevant.

Explanation

As we can't discriminate between the indistinguishable apples, the order of the apples isn't crucial, thus we'll apply the notion of a combination.

Let's start by picking three apples of each type. Then we must select three apples from the remaining three boxes.

\(\begin{array}{l}{\rm{n = 3}}\\{\rm{r = 3}}\end{array}\)

Since repetition is allowed:

\(\begin{array}{c}{\rm{C(n + r - 1,r) = C(3 + 3 - 1,3)}}\\{\rm{ = C(5,3) = }}\frac{{{\rm{5!}}}}{{{\rm{3!(5 - 3)!}}}}\\{\rm{ = }}\frac{{{\rm{5!}}}}{{{\rm{3!2!}}}}\\{\rm{ = 10}}\end{array}\)

Hence, there are \({\rm{10}}\)ways to choose a dozen apples.