8 person purchased 6 types of beverages, each person purchased one type. In total, how many different ways of buying these beverages? 8 person purchased 6 types of beverages, each person purchased one type. In total, how many different ways of buying these beverages? I would say that as each of the 8 eight person has 6 choices, it would be > 8^6 but I am not sure.

As you said, each person has $6$ choices. And there are $8$ persons. So the answer should be $6^8$. This directly follows from Rule of product.