you wrongly prohibit repetitions.
the question stem: exactly one of the boxes should be on the fifth floor and other four could be placed at any other stories?
No conditon that exactly one box should lie on a floor, considering the other 4 boxes. They could lie together on each floor, for example.
Moreover, 5! means five boxes on five places, but you have 5 boxes and 6 floors.
But I also made a mistake: exactly one of the boxes should be on the fifth floor -- we should take 1 out of 5; 5C1=5. The other four give 5^4 ways to be placed.
Agree with you.... my mistake