A logical puzzle about a pet shop

I am an undergraduate student studying languages and give private lessons to secondary school students to help them learn English. Recently one of them suddenly asked me during a lesson whether I could help him solve a logical problem given to him by his math teacher as a part of an assignment. I agreed, but only managed to come up with a half-intuitive answer. Seeing my confusion, the boy told me to get on with the English lesson. I’m very curious whether I told him the right answer, and I’m curious how to solve the problem.

The problem is this: There’s a pet shop, and there are 100 pets inside it. Those pets are hamsters, ferrets, chinchillas, and guinea pigs. That is, four mutually exclusive kinds of pets. You don’t know how many pets of each kind are there in the shop, but, as stated above, you know that the total number is 100. And you also know one more thing: if you select any 85 pets in this shop, then there will invariably be at least one hamster, at least one ferret, at least one chinchilla, and at least one guinea pig among the selected pets. And here is the question: given this information, how many pets must you select at random in this shop to ensure that the selected pets are of at least three different kinds?

The half-intuitive answer I gave the boy is 69, but I’m highly unsure whether I’m not missing anything, because there are a multitude of possibilities within the restrictions specified in the problem. The logic needed to solve the problem seems very convoluted, and I am lost in my chaotic thoughts.

Can you come up with a logically clear solution and tell me what the right answer is?