In how many ways can we distribute 2 types of gifts?
The problem: In how many ways can we distribute 2 types of gifts, m of the first kind and n of the second to k kids, if there can be kids with no gifts?
From the stars and bars method i know that you can distribute m objects to k boxes in ways. So in my case i can distribute m gifts to k kids in ways, same for n gifts i can distribute them in ways. So now if we have to distribute m and n gifts we can first distribute m gifts in ways, then n gifts in ways, so in total we have:
Is my reasoning correct?
What about when we have to give at least 1 gift to each kid, can we do that in