Question

The set $S=\{1,2,3,\dots .,12\}$ is to be partitioned into three sets $A,B,C$ of equal size, thus $A\cup B\cup C=S$, $A\cap B=B\cap C=A\cap C=\varphi$. Thus number of ways to partition $S$ is

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

Answer 1

Answer: (D)

$\frac{12!}{(4!{)}^3}$

Since $S\{1,2,3,...,12\}$ is to be partitioned into the sets $A,B,C$ of equal size,

i.e., each set of $4$ elements

It is given that $A\cap B=B\cap C=A\cap C=\varphi$

So, in each set all the elements are different.

So, the required number of ways $={}^{12}{C}_4{\times }^8{C}_4{\times }^4{C}_4$ $=\frac{12!}{{(4!)}^3}$