I have tried to figure this out and I cannot. The professor gave us an answer of 13,536 but I do not see any way in which he got to his answer. Any help would be greatly appreciated.
A certain classroom has two rows of seats. The front row contains 8 seats and the back row contains 10 seats. How many ways are there to seat 15 students if a certain group of 4 or them refuses to sit in the front row?
There are $10\times9\times8\times7=5040$ ways to seat those four that want to be in the first row. Then there are 14 seats and 11 students left, and there are $14!/(14-11)!=14529715200$ ways to seat them. The total number is the product of these two, namely 73229764608000. This is somewhat larger than your professor's answer, though...
I assumed here that the students are identifiable, which (to me) seems to be the only reasonable interpretation since they have personal preferences.