We can think of the problem as sampling with replacement where order matters.
There are possible sequences of outcomes. We are interested in the cases where each face of the die is rolled exactly times. Since each sequence is equally likely, we can use the naive definition of probability.
There are ways to select the dice that fall on a . Then, ways to select the dice falling on a , falling on a , falling on a , falling on a and finally, falling on a .
Thus, the desired probability is
Alternatively, imagining the sample space to be a digit long sequence of , we want the cases in which each of numbers appear exactly five times. There are ways to arrange such a sequence. Hence, the probability is