Question

How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD?

Short answer

There are three ways to arrange eight indistinguishable objects into five indistinguishable boxes

Step-by-step solution

Concept Introduction

Counting is the act of determining the quantity or total number of objects in a set or a group in mathematics. To put it another way, to count is to say numbers in sequence while giving a value to an item in a group on a one-to-one basis. Objects are counted using counting numbers.

How many ways are there to pack eight identical DVDs into five indistinguishable 

We're trying to figure out how many different methods there are to arrange eight indistinguishable things (identical DVDs) into five indistinguishable boxes.

We won't be able to solve this problem using a formula, therefore we'll have to figure out every conceivable combination of the number of objects in the boxes.

At least one element must be present in each of the five boxes.

\[\begin{array}{*{20}{r}}{{\rm{4,1,1,1,1}}}&{{\rm{ (4 objects in one box, 1 in each of the four other boxes) }}}\\{{\rm{3,2,1,1,1}}}&{{\rm{ (3 objects in one box, 2 in another and 1 in each of the remaining boxes) }}}\\{{\rm{2,2,2,1,1}}}&{{\rm{ (2 objects each in three boxes, 1 in each of the remaining boxes) }}}\end{array}\]

We then see that there are three ways to arrange eight indistinguishable objects into five indistinguishable boxes, with at least one element in each box.