Question: $20$ distinct students are to be placed into four distinct dorms named: A, B, C, D. In how many ways can they be assigned to the four dorms, with the restriction that each dorm needs to have at least one student?
My attempt: The question says that each dorms must have at least one student. So, my first attempt is since there are four dorms, then the first dorm has $20$ choices to take in one of the $20$ students, and the second dorms then has $19$ choices to take in one students. The third dorms has 18 choices, and the fourth has $17$ choices. Now, each dorms has one student, and that leaves $16$ students left who may enter any one of the dorms, so the arrangement is $16^4$. So there are $20 \times 19 \times 18 \times 17 \times 16^4$ arrangements.
However, it seems that I can first distribute the 16 students into the dorms with $16^4$ arrangements, then distribute the remaining 4 students with $4!$ arrangements, so that each dorms has at least one student in them. So the total arrangement is $16^4 \times 4! < 20 \times 19 \times 18 \times 17 \times 16^4$. This don't seem right.
I appreciate it very much if anyone could help me out on this problem. Thank you.