2. At a cactus lovers' meeting, 80 cactus enthusiasts presented their collections, each consisting of cacti of different species. It turned out that no species of cactus is present in all collections at once, but any 15 people have cacti of the same species. What is the smallest total number of cactus species that can be in all the collections?
Answer: 16.

Solution. We will show that 16 cacti were possible. Number the cacti from 1 to 16. Let the 1st cactus lover have all cacti except the first; the 2nd cactus lover have all except the second cactus; the 15th cactus lover have all except the fifteenth cactus; and the cactus lovers from the 16th to the 80th have all cacti except the sixteenth. Then, any 15 cactus lovers will have a common cactus.

Now, we will establish that they must have more than 15 cacti. Suppose the opposite: let there be a total of $k \leqslant 15$ cacti. Number the cacti from 1 to $k$. For the cactus with number $i$, there exists a cactus lover $A_{i}$ who does not have it. But then, for the cactus lovers $A_{1}, A_{2}, \ldots, A_{k}$, there is no cactus that all of them have. And, even more so, there is no such cactus if we add a few more cactus lovers to them so that their number becomes 15. Contradiction.