Number of Arrangements of n Objects of m Types/Examples

Examples of Use of Number of Arrangements of $n$ Objects of $m$ Types

$2$ Types

Let $S$ be a collection of $n$ objects, consisting of:

$p$ objects of one type

$q$ objects of another type.

The total number $N$ of different arrangements of $S$ is given by:

$N = \dfrac {n!} {p! \, q!}$

$3$ Types

Let $S$ be a collection of $\paren {p + q + r}$ objects.

Let $S$ need to be partitioned into $3$ subsets of size $p$, $q$ and $r$ such that $p \ne q$, $q \ne r$ and $r \ne p$.

The total number $N$ of ways this can be done is:

$N = \dfrac {\paren {p + q + r}!} {p! \, q! \, r!}$

Set of $3 p$ Objects of $3$ Equal Sized Subsets

Let $S$ be a collection of $3 p$ objects.

Let $S$ need to be partitioned into $3$ subsets of size $p$.

The total number $N$ of ways this can be done is:

$N = \dfrac {\paren {3 p}!} {\paren {p!}^3 \times 3!}$

Letters in added

Let $N$ be the number of different arrangements of the letters in the word $\texttt{added}$.

Then:

$N = 20$

$6$ people in $3$ pairs

Let $N$ be the number of ways $6$ people can be partitioned into $3$ (unordered) pairs.

Then:

$N = 15$

$10$ people in $3$ groups of sizes $5$, $3$ and $2$

Let $N$ be the number of ways $10$ people can be partitioned into $3$ sets: one with $5$, one with $3$ and one with $2$ people.

Then:

$N = 2520$