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The setting

I know how to calculate the number of combinations for any given sum and side of a dice but i wanted to caculate the number of combination for any given sum with a dice starting not a one but at two and still ending at six.

I visualized the problem as a set problem with a venn diagram.

I first computed the number of combination giving me the sum T using t six sided dices ( = the common formula).

Then i realized that any combinations including a one and giving me a sum T had to have an intermediary sum of T-1 with t-1 dices. So I first computed the number of combinations giving me T with t dices and then I deduced the number of computations giving me T-1 with t-1 dices :


$$P(t,s,T)=( \sum_{k=0}^{\lfloor{\frac{T-t}{s}}\rfloor} (-1)^k * \frac{n!}{(n-k)!k!}\frac{(T-s*k-1)!}{(T-s*k-n)!*(n-1)!}-?*\sum_{k=0}^{\lfloor{\frac{T-t}{s}}\rfloor} (-1)^k * \frac{n!}{(n-k)!k!}\frac{(T-s*k-2)!}{(T-1-s*k-n)!*(n-1)!}) *(\frac{1}{s})^t$$


Where:
=number of rolls
=number of cards
=total value
?= what i need help to find

The problem

The difficulty arise when i have to multiply the number of combinations giving me T-1 with t-1 side to find the number of combinations they birthed with t dices.

I m really stuck and I'm now at the point where I have cut sheet of paper in tiny bits to visualize all of this and my desk is an absolute mess.

Really a huge thanks for any help !!

Yellow Dot
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  • Duplicate problem already answered. See this answer. One difficulty is that I am construing the $n$ dice to be distinguishable. If that is not your intent, then the linked answer does not apply. – user2661923 Jun 15 '22 at 00:51
  • Re previous comment, you also have the trivial difference that the lower bound is $(2)$ for each die, rather than $(0)$. However, that difference is easily adjusted for using the methods in the linked answer. – user2661923 Jun 15 '22 at 00:55
  • Another problem with my linked answer is that it does not use recursion, in any way. From your posting, it seems as if you are looking for a recursion based approach. For these types of problems, assuming that the dice are distinguishable, I have seen Stars and Bars theory used, or generating functions used. I have never seen recursion used. – user2661923 Jun 15 '22 at 01:01

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