Dice Combinations for a Particular Sum

[Chronos0]

Okay, here is a problem that has been bugging me:

Say that you have n dice that each have d sides, numbered 1 through m. How many different dice combinations, out of a total of n x d possible, are there where the sum of all of the dice is equal to r, for particular n, d, and r?

I'm not necessarily looking for a general solution, though one would be great. But I would like to at least be able to compute the answer for particular sets of input.

So far, I believe I have found that answering the main question is equivalent to determining the number of (r−n) -combinations in a multiset with the form

{{k₁, ... , k_n}, {(k₁, d−1), ... , (k_n, d−1)}}

Basically, starting with each die at one and counting the number of ways I can build up to the total, r, without exceeding d in any of them.

I can find a solution where d = ∞ easily enough, but I am not sure how to eliminate the illegal combinations from that using Inclusion-Exclusion.

If someone could help me finish my solution or propose an alternate, easier solution, I would be very grateful.


(Added:)
Basically, if we are using two six-sided dice, then there are six ways to add up to seven:

1+6, 2+5, 3+4, 4+3, 5+2, and 6+1.

And five ways to add up to eight:

2+6, 3+5, 4+4, 5+3, and 6+8.

Et cetera.

 

[Valkarie]

A possible solution is to count all of the possible combinations (n x d) and then subtract out the combinations that do not add up to r. You can use Inclusion-Exclusion to do this, by breaking down each number that does not add up to r into its components and subtracting out those components. For example, if you have two six-sided dice and are trying to find the combinations that add up to seven, you would first count all the possible combinations (12). Then, subtract out the combinations that add up to 5, 6, 8, 9, 10, 11, and 12. This can be done by subtracting out the combinations that make up those numbers, such as 3+4 for 7, 4+3 for 7, 2+5 for 7, etc. In the end, this should give you an answer of 6, which corresponds to the number of combinations that add up to seven with two six-sided dice.