Combinations help with last question

[Eagerissac]

I got a, b, and I will put the questions there for context but can someone please explain how to do c? I'm lost as to how to approach the last question. Like I got an answer but my teacher said it's not right and I don't know how else to do it.

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

a. Assuming my friend determined this entirely by testing, how many experiments did my friend have to perform?

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

c. Suppose that an impatient novice wants to repeat my friends discovery using fewer experiments. This novice is wiIIing to take 5 drugs at a time. What is the fewest number of experiments this novice can perform so that for any triple Di,Dj,Dk∈{D1,…,D21}, at Ieast one of this novice's experiments includes Di, Dj, and Dk?

21C3/5C3 is what I got but it's apparently wrong

 

[Deleted member 4993]

I got a, b, and I will put the questions there for context but can someone please explain how to do c? I'm lost as to how to approach the last question. Like I got an answer but my teacher said it's not right and I don't know how else to do it.

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

a. Assuming my friend determined this entirely by testing, how many experiments did my friend have to perform?

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

c. Suppose that an impatient novice wants to repeat my friends discovery using fewer experiments. This novice is wiIIing to take 5 drugs at a time. What is the fewest number of experiments this novice can perform so that for any triple Di,Dj,Dk∈{D1,…,D21}, at Ieast one of this novice's experiments includes Di, Dj, and Dk?

21C3/5C3 is what I got but it's apparently wrong

Please share your for (a) and (b) - that will give us an idea about where we would need to start explaining.