Category:Number of Arrangements of n Objects of m Types

This category contains pages concerning Number of Arrangements of n Objects of m Types:

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:

$N = \dbinom n {k_1, k_2, \ldots, k_m} = \dfrac {n!} {k_1! \, k_2! \cdots k_m!}$