Question

If the set $S=\{1,2,3,\cdots ,12\}$ is to be partitioned into three sets $A,B,C$ of equal size such that $A\cup B\cup C=S,A\cap B=B\cap C=A\cap C=\varphi$ then the number of ways of partitioning $S$ is :

• A. $\frac{12!}{(3!{)}^4}$
• B. $\frac{12!}{3!(4!{)}^3}$
• C. $\frac{12!}{3!(3!{)}^4}$
• D. $\frac{12!}{(4!{)}^3}$

Answer 1

The correct option is D $\frac{12!}{(4!{)}^3}$

The above problem can be stated as distributing $12$ object in three different boxes
Hence the required number of ways
$=\frac{12!}{(4!{)}^3\cdot 3!}\cdot 3!=\frac{12!}{(4!{)}^3}$