How many ways of moving $6$ wine bottles from the cellar to the fridge? I have a seemingly easy question about combinatorics but I seem to be missing something for the correct answer. The questions is as follows: > You are organizing a party and your guests want to drink wine. You have 3 different brands in your cellar. Per brand you have 12 bottles. In preparation of the party you want to move some bottles to your fridge. However your fridge only fits 6 bottles. How many possibilities do you have to choose 6 bottles to put in your fridge? The answer is given as 28 possibilities but I can't seem to figure out why that's the solution. I tried so many different approaches already but I think I'm missing a catch somewhere. Any help that would even lead me in the right direction would be appreciated

I don’t think it’s very easy.

There are 3 ways of choosing only one brand (choose a brand, take 6 bottles).

There are 15 ways of choosing 2 brands:

3 ways of choosing which brand not to include, multiplied by 5 ways of choosing 6 bottles from the other two brands (since you can take 1-5 of one brand, and 6 minus that for the other).

There are 10 ways of choosing 6 bottles if you include all three brands:

You have to have at least one bottle from each, but you can choose the other three bottles without constraint, so for this sub problem:

3 ways of choosing all three bottles from the same brand, 1 way of choosing all three brands (1 bottle of each), and 6 ways of choosing the three bottles from two brands: 3 ways to choose which not to include, then two choices for which to take 2 of.

Since these are mutually exclusive, there wasn’t any double counting, so there are a total of $3+15+10=28$ ways of choosing the 6 bottles.