Number of Arrangements of n Objects of m Types

Theorem
Let $S$ be a collection of $n$ objects.

Let these $n$ objects be of $m$ different types, as follows:

Let there be:
 * $k_1$ objects of type $1$
 * $k_2$ objects of type $2$
 * $\cdots$
 * $k_m$ objects of type $m$

such that:
 * for each $j \in \set {1, 2, \ldots, m}$, all objects of type $j$ are indistinguishable from each other
 * $k_1 + k_2 + \cdots + k_m = n$

Then the total number $N$ of different arrangements of $S$ is given by the multinomial coefficient:
 * $N = \dbinom n {k_1, k_2, \ldots, k_m} = \dfrac {n!} {k_1! \, k_2! \cdots k_m!}$

Proof
Let $N$ be the number of different arrangements of $S$.

First suppose that all $n$ objects are distinct one from another.

Then from Number of Permutations of All Elements:
 * $N = n!$

Now suppose that $k_j$ elements of $S$ are indistinguishable from each other.

From Number of Permutations of All Elements, there are $k_j!$ different arrangements of those $k_j$ elements.

Hence the $n!$ arrangements of $S$ can each be grouped into $\dfrac {n!} {k_j!}$ partitions, each with $k_j$ elements, such that all the arrangements in each partition are indistinguishable from each other.

This applies for all $m$ types of objects.

The result follows.