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

Question:

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

Solution:

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. 

$\therefore$ Total number of ways $=\frac{{ }^{12} C_{4} \times{ }^{8} C_{4} \times{ }^{4} C_{4}}{3 !}=\frac{12 ! \times 8 !}{4 ! \times 8 ! \times 4 ! \times 4 ! \times 3 !}=\frac{12 !}{(4 !)^{3} \times 3 !}$

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