Question

Write the number of ways in which $12$ boys may be divided into three groups of $4$ boys each.

Answer 1

Finding the number of such possible arrangements

$12$ boys are to be divided in $3$ groups. Now, $4$ boys can be chosen out of $12$ boys by
${}^{12}{C}_4\times {}^8{C}_4\times {}^4{C}_4$ ways.

These $3$ groups of equal size can be arranged in $3!$ Ways which must be excluded. Therefore, total number of ways

$=\frac{{}^{12}{C}_4\times {}^8{C}_4\times {}^4{C}_4}{3!}$

Hence, the total number of ways

$=\frac{12!\times 8!}{4!\times 8!\times 4!\times 4!\times 3!}=\frac{12!}{(4!{)}^3\times 3!}$