Question

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

Answer: (C)

Set $S=\{1,2,3, \ldots, 20\}$ 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
$={ }^{20} C_{5} \times{ }^{15} C_{5} \times{ }^{10} C_{5} \times{ }^{5} C_{5}$
$=\frac{20 !}{15 ! \times 5 !} \times \frac{15 !}{10 ! \times 5 !} \times \frac{10 !}{5 ! \times 5 !} \times 1$
$=\frac{(20 !)}{(5 !)^{4}}$