Question
The set $$S = \left\{ {1,2,3,......,12} \right\}$$ is to be partitioned into three sets $$A, B, C$$ of equal size. Thus $$A \cup B \cup C = S,$$ $$A \cap B = B \cap C = A \cap C = \phi .$$ The number of ways to partition $$S$$ is
A.
$$\frac{{12!}}{{{{\left( {4!} \right)}^3}}}$$
B.
$$\frac{{12!}}{{{{\left( {4!} \right)}^4}}}$$
C.
$$\frac{{12!}}{{3!{{\left( {4!} \right)}^3}}}$$
D.
$$\frac{{12!}}{{3!{{\left( {4!} \right)}^4}}}$$
Answer :
$$\frac{{12!}}{{{{\left( {4!} \right)}^3}}}$$
Solution :
$$\eqalign{
& {\text{Set }}S = \left\{ {1,2,3,......,12} \right\} \cr
& A \cup B \cup C = S,A \cap B = B \cap C = A \cap C = \phi . \cr} $$
∴ The number of ways to partition
$$ = {\,^{12}}{C_4} \times {\,^8}{C_4} \times {\,^4}{C_4} = \frac{{12!}}{{4!8!}} \times \frac{{8!}}{{4!4!}} \times \frac{{4!}}{{4!0!}} = \frac{{12!}}{{{{\left( {4!} \right)}^3}}}$$