The most likely sum of any roll of n unbiased s-sided dice is given by:
$ P (n, s) = \frac{1}{2} n ( s + 1 )$
This tells us, of course, that for an odd number n, the most likely sum will be fractional; that is, two sums will be equally likely.
Is there a simple, closed formula that would determine the probability of that most likely sum? It strikes me that such a formula should exist, since any particular sum has a definite and predictable number of summands, based upon $n$ and $s$. And since the total number of permutations is $s^n$, the question becomes one of calculating the number of ways the most likely number can be achieved with those dice.
Thank you.