Question

$S=\{1,2,3,...20\}$ is to be partitioned into four sets $A,B,C$ and $D$ of equal size. The number ways it can be done, is

• A. $\frac{20!}{4!\times 5!}$
• B. $\frac{20!}{4^5}$
• C. $\frac{20!}{(5!{)}^4}$
• D. $\frac{20!}{(4!{)}^5}$

Answer 1

The correct option is C $\frac{20!}{(5!{)}^4}$

Set $S=\{1,2,3,...20\}$ is to be partitioned into four sets of equal size i.e., having $5-5$ numbers.
The number of ways in which it can be done
${=}^{20}{C}_5{\times }^{15}{C}_5{\times }^{10}{C}_5{\times }^5{C}_5$
$=\frac{20!}{15!\times 5!}\times \frac{15!}{10!\times 5!}\times \frac{10!}{5!\times 5!}\times 1=\frac{(20!)}{(5!{)}^4}$.