This is basically Example 2.5 from Elements of Discrete Mathematics by C L Liu. I'm trying to break down the problem rather than just plug in the formula and arrive at the result, which is 125. Theoretically the number of ways to arrange r distinct objects into n boxes or bins is $n^r$. And so by this formula, the number of ways to schedule three exams in a 5-day period is $5^3 = 125$ provided that each day can hold as many exams as we wish.
So suppose if a, b, c are our exams that I would like to arrange for day 1. How can we describe them in writing? These can be abc, acb, bac, bca, cab, cba. But is that all? How does one prove the formula $n^r$?
The first exam has a total of $5$ choices. Similarly the second exam and third exam have $5$ choices.
So the answer is $5^3=125$ ways.
Now lets consider the general problem of placing $r$ objects into $n$ distinct bins.
Consider the first object. This can be placed into any of the $n$ bins. So it has $n$ choices. Similarly, since there is no restriction of number of objects in a bin, we again have $n$ choices. Hence the formula is $n^r$.