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?