nt.number theory - Permutations with identical objects

Note: I tried asking this on math.stackexchange (here), but didn't really receive an answer - so I figured this might be the right place.

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How can I find the number of k-permutations of n objects, where there are x types of objects, and r₁, r₂, r₃ ... r_x give the number of each type of object?

Example:

I have 20 letters from the alphabet. There are some duplicates - 4 of them are a, 5 of them are b, 8 of them are c, and 3 are d. How many unique 15-letter permutations can I make?

In the example:

n = 20
k = 15
x = 4
r₁ = 4, r₂ = 5, r₃ = 8, r₄ = 3

Furthermore, if there isn't a straightforward solution: how efficiently can this problem be solved?