Number of Arrangements of n Objects of m Types/Examples/3p Objects into 3 Equal Sized Subsets

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

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:

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

From Number of Arrangements of $n$ Objects of $3$ Types, we can partition $S$ into $3$ subsets in $\dfrac {\paren {3 p}!} {\paren {p!}^3}$ different ways.

However, each of these subsets is of the same size $p$.

Hence they cannot be counted as different.

So we have counted $3!$ ways which are apparently all different, but are in fact the same.

Hence we need to divide our total by $3!$

Hence $N$ can be given by:

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

$\blacksquare$

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: The number of ways of dividing $3 p$ things into three equal groups each containing $p$ things