The general problem I'm trying to solve is the probability of rolling a total t on n s-sided dice. A good chunk of the problem is easy enough, but where I run into difficulty is this:(adsbygoogle = window.adsbygoogle || []).push({});

How many combinations of dice will yield a sum total of t? Because the number set is limited, [tex]{a \choose n-1}[/tex] (where [tex]a={n(s+1) \over 2} - \left|{n(s+1) \over 2} - t\right|[/tex]) no longer works when [tex]n+s \leq t \leq (n-1)s[/tex]. It is this region in the middle that interests me. Enumerating all combinations could be time-consuming, and, I expect, is entirely unnecessary. Is there a known formula for computing these numbers?

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# Sum of n elements of a finite set of integers, 1 through s

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