12. PQR Entertainment wishes to divide their popular idol group PRIME, which consists of seven members, into three sub-units - PRIME-P, PRIME-Q, and PRIME-R - with each of these sub-units consisting of either two or three members. In how many different ways can they do this, if each member must belong to exactly one sub-unit?
Answer: 630
Solution: Note that the only way to do this is to divide PRIME into two sub-units of two members and one sub-unit of three members. There are $\binom{7}{3}\binom{4}{2}\binom{2}{2}=210$ ways to do this. Since there are 3 ways to choose which unit will have three members, our final answer is $210 \cdot 3=630$.