A certain friend of mine has spent the better part of a lifetime testing recreational drugs. After thorough testing, this friend has identified 20 recreational drugs D1,…,D20 and determined (experimentally) that any 3 of these drugs can be taken simultaneously with no adverse effects.

1. Suppose my friend survives the experience and D21 makes it onto the list. My friend takes scrupulous notes about all experiments and notices something peculiar about the answers to the preceding two questions. What combinatorial identity did my friend just discover?

2.A new designer drug called D21 has just hit the streets and my friend wants to know if D21 can be added to their list. That is, can any triple of D1,…,D21 be safely taken together? How many additional experiments does my friend need to determine this?