Number of Arrangements of n Objects of m Types/Examples/2 Types
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Example of Use of Number of Arrangements of $n$ Objects of $m$ Types
Let $S$ be a collection of $n$ objects, consisting of:
The total number $N$ of different arrangements of $S$ is given by:
- $N = \dfrac {n!} {p! \, q!}$
Proof
An arbitrary arrangement can be made into $p!$ arrangements if the $p$ objects of type $1$ are replaced by $p$ objects that are all different.
Similarly, an arbitrary arrangement can be made into $q!$ arrangements if the $q$ objects of type $2$ are replaced by $q$ objects that are all different.
The total number of arrangements, if all objects are different, would be $n!$
Hence:
- $n! = N \times p! \times q!$
Hence the result.
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: The number of ways of arranging $q$ things in line if $p$ are alike of one kind and $q$ are alike of another kind