TITLE: How many ways $12$ persons may be divided into three groups of $4$ persons each?
QUESTION [8 upvotes]: How many ways $12$ persons may be divided into three groups of $4$ persons each?

I think the answer should be $\frac{12!}{(4!)^3}$ but the suggested correct answer is $5775$, could anybody explain where I am going wrong?

REPLY [2 votes]: From $12$ persons , $4$ persons can be chosen in $\binom{12}{4}$ ways. From rest $(12-4)=8 $ persons 4 persons can be chosen in $\binom{8}{4}$ ways.Remaining $4$ will form the third group.Thus $\binom{12}{4} \binom{8}{4}$ are the ways for form three groups of $4$ persons out of $12$ persons.But we have introduced order in group formation. Now three groups can be permuted $3!$ ways and they are all same.Hence correct number of ways $= \frac{\binom{12}{4} \binom{8}{4}}{3!}=5775 $