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

[Nessalc]

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:

How many combinations of dice will yield a sum total of t? Because the number set is limited, $${a \choose n-1}$$ (where $$a={n(s+1) \over 2} - \left|{n(s+1) \over 2} - t\right|$$) no longer works when $$n+s \leq t \leq (n-1)s$$. 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?

 

[zli034]

sum of the m rolls diverges as m gets large. but we can use standardize to make it converge to standard normal distribution. this the weak law of large number says.

 

[Nessalc]

I'm not sure I understand what you're saying. The total t must be in the range [n, ns]. I am aware of the idea that it would probably converge to a normal distribution, but for $${1 \over {\sqrt{2\pi\sigma^2}}} e^{-{{\left((x-{n(s+1) \over 2})-\mu\right)^2} \over {2\sigma^2}}}$$, what would $$\sigma$$ and $$\mu$$ be in terms of n, s and t?

 

[awkward]

See this page:

http://mathworld.wolfram.com/Dice.html

 

[Nessalc]

That's exactly what I was looking for, thanks!