Question

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

Answer 1

Number of groups in which 12 boys are to be divided = 3
Now, 4 boys can be chosen out of 12 boys in $\left(C_4{\times }^8{C}_4{\times }^4{C^{12}}_4\right)$ ways.
These groups can be arranged in 3! ways.
∴ Total number of ways = $\frac{{}^{12}{C}_4\times {}^8{C}_4\times {}^4{C}_4}{3!}=\frac{12!\times 8!}{4!\times 8!\times 4!\times 4!\times 3!}=\frac{12!}{{\left(4!\right)}^3\times 3!}$