It's Friday, and I'm tired, so there possibly a trivial solution that I am overlooking ...
The problem is the following: I have ($n=2$) two kinds of things, say Apples and Bananas, and I choose $k$ of these, not minding the order in which I take them.
What is the possible number of combinations, given that I only have $\alpha$ Apples and $\beta$ Bananas? ($\alpha+\beta>k$, but typically $\alpha<=k$ or $\beta<=k$ or both). And how many solutions are there that include exactly $\gamma<=\alpha$ Apples?
(I think that without the limits the solution would be an application of the multiset-number/stars-and-bars, with n and k. However, these solutions assume a unlimited supply of fruit, and I cannot see a way to adapt that these results to my limits.)