Question

How many ways are there to put n identical objects into m distinct containers so that no container is empty.

Short answer

There are \({\rm{C(n - 1,n - m)}}\) ways to put n identical objects into m distinct containers so that no container is empty.

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

Let \({\rm{n}}\)Similar objects must be placed in \({\rm{m}}\) different containers.

Now, start by putting one item in each container; after that, we'll have \({\rm{n - m}}\) identical objects.

\(\begin{array}{l}{{\rm{n}}^{\rm{*}}}{\rm{ = m}}\\{\rm{r = n - m}}\end{array}\)

Because the order of the things is irrelevant (as they are all the same), we must apply the concept of a combination.

Because more than one thing can be in a container, repetition is permitted:

\({\rm{C(n + (n - m) - 1,n - m) = C(n - 1,n - m)}}\).

Hence, there are \({\rm{C(n - 1,n - m)}}\) ways to put n identical objects into m distinct containers so that no container is empty.