Well, in your previous brute force attempt, you listed more than 10 possibilities, so we know for sure that can't be the answer. This suggests that simply adding the number of possibilities is not correct. But let's investigate why that might be, and see if we can't stumble upon what
is the correct answer. Suppose for a moment there were only two slots and we could choose four sodas for the first slot and three for the second slot. Right away we can begin listing the possibilities for the first slot (again, labeling the sodas A, B, C, and D):
Here [?] denotes a "wildcard" - that particular slot can be anything. We know that the second slot has three possibilities, so we can fill in the missing information:
In other words,
for each of the four sodas we put in the first slot, we have three choices for the second slot. The words I've bolded are the very important bit, as thinking about what those words mean with reference to word problems should give you an inclination as to why it makes intuitive sense to multiply rather than add.