Combinatorics: r-Combinations on a Multiset

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Homework Statement


How many r-combinations are there of a multiset S = {1*a1, infinity*a2,...,infinity*ak}?


Homework Equations


The number of r-combinations on a multiset with k objects and infinite repetition number: (r+k-1) choose (r), or (r+k-1) choose (k-1).
The number of r-combinations on a multiset with k objects and finite repetition number is 1-to-1 with the number of positive integral solutions to x1 + x2 + ... + xk = r.


The Attempt at a Solution


I'm not sure how to solve this problem since there is a mix of infinite and finite supplies of k objects.

The answer is (n+k-2) choose (k-2) + (n+k-3) choose (k-2)...
 
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