Discrete Mathematics - Ice Cream random samples

Question

How would you solve the following problem with Discrete Mathematics, and what is the answer?

Suppose there are 5 different types of ice cream you like. How many random samples ice cream must be eaten to guarantee that you have had at least 7 samples of one type?

Answer 1

HINT: You can eat $30$ samples without getting $7$ of the same type; how? What happens if you eat more than that?

Answer 2

Hint: If you have eaten $(5)(6)$ samples, you might not have eaten $7$ of any flavour, since you might have eaten $6$ of each flavour. But anything over that $\dots$. In your answer, you might be expected to mention and use the Pigeonhole Principle.