Equal Team Allocation

This is a multinomial coefficient splitting $12$ people into groups of sizes $4,4,4$.

Choose the first team, then the second from what remains; the third is determined.

1. 1. Apply the multinomial coefficient

   Choosing an ordered partition into labeled groups of sizes $4,4,4$ from $12$ distinct people uses the multinomial coefficient.

   $$\left(\genfrac{}{}{0ex}{}{12}{4,4,4}\right)=\frac{12!}{4!4!4!}$$

2. 2. Evaluate

   Computing the factorial ratio gives the final count.

   $$\frac{12!}{(4!{)}^3}=34650$$

Answer: 12!/(4!)^3