Dice with k-sides

A standard 6 sided die rolls the numbers 1 through 6 with equal probability. Likewise, a k-sided die rolls the numbers from 1 through k with equal probability.

Let D(S, N, k) equal the number of ways to roll a total of S with N k-sided dice. D(3, 2, 6) = 2 because you can roll a 3 with a (1, 2) or a (2, 1).

Let T = D(20, 7, 12) + D(31, 6, 4) + D(15, 3, 7) + D(111, 17, 7) + D (17, 3, 57) + D(1, 2, 2) + D(10, 17, 12) + D(9, 3, 0).

What are the last three digits of T?

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