I've got a problem where I have `N` items with up to `D` duplicates for each item.  I want to know how many unique sets of `K` of the input items I will have given my inputs.

I've got an application that actually generates these unique permutations and works on them, but I'd like to understand how I can compute the number of sets I'll have across various inputs without computing the entire result.

Example (in R):

    N <- 19
    K <- 4
    # Implied D = 3 by just duplicating it in-place three times.

    a <- append(1:N, append(1:N, 1:N))
    b <- unique(gtools::permutations(length(a), K, a, set=FALSE))

`nrow(b)` in this case will be `130,302`.

This is slow and inelegant.  Can someone help me do this with actual math?