For this problem the first you have to get the assumption for floor distribution relative to the passengers.
Now you have to look at the possibilities regarding passenger arrangements this is equivalent to a multinomial distribution with n = 5 trials and eight possibilities which is a generalized version of the binomial where you sample without replacement (in other words once the person gets off the elevator he is not put back in the sample), and the probability of each event will be 1/8 since all arrangements have the same probability.
Multinomial distribution - Wikipedia, the free encyclopedia
So basically you have this PDF and you need to find the probabilities that no two more passengers leave the same floor. This corresponds to having four of the people on board selecting a separate floor and the last person having the choice of selecting one from the remaining four, but you need to allow for all possibilities.
Now you can evaluate this probability in many ways. One way is to consider the complement of the situation and do one minus that and the other is to get the actual probability.
I'll get your response before we continue but there are many ways of evaluating things in mathematics.
Personally for this I might resort to using a computer by running a small program but I'm guessing you need to practice for some kind of entrance exam so this might not be appropriate for your situation.