How can I find the number of $k$-permutations of $n$ objects, where there are $x$ types of objects, and $r_1, r_2, r_3, \cdots , r_x$ give the number of each type of object?
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:
$n = 20$
$k = 15$
$x = 4$
$r_1 = 4 \quad r_2 = 5 \quad r_3 = 8 \quad r_4 = 3$
I was originally going to solve this problem in order to solve a simpler problem, but I managed to find a simpler solution to that problem instead. Now I'm still looking for the solution to this more general problem out of interest.
I've done some more work on this problem but haven't really come up with anything useful. Intuition tells me that as Douglas suggests below there will probably not be an easy solution. However, I haven't been able to prove that for sure - does anyone else have any ideas?
I've now re-asked this question (here) on MO.