A travel agency organized four optional tours of Sicily for a group of tourists. Each tour had a participation rate of 80%. What's the smallest possible percentage of tourists who took part in all four tours? So first, let T1, T2, T3, and T4 be the sets of people who participated in the four tours. We want to minimize the size of their intersection, meaning the people who participated in tour 1 and tour 2 and tour 3 and tour 4. And let's say that the size is measured as the percent of the total. Well, if we want to minimize this set, we could also maximize its complement, which is T1 complement, union, T2 complement, so on and so forth. Well, this is the people who missed at least one tour, which is what we want to maximize. Well, we know that the size of a union is at most the sum of the sizes of these sets inside of here. So we know that this is at least T1 complement plus T2 complement, so on and so forth. Well, each tour had a participation rate of 80%. That means 20% missed the tours. That means each one of these four sets has size 20%, so they add up to 0.8. Well, that means that this intersection is at most 1 minus this guy right here. So the intersection is at most 20%. So now the question is, can we get, you know, this thing to be equal at some point? And the answer is yes, we can. Let's do it with five people. Let's just do person 1, person 2, person 3, person 4, person 5. And so we need one fifth of them to go on all the tours. So let's just make this guy go on all the tours. And then for each other person, we just choose one tour to miss. And this does the trick. So our answer is 20%.

Sicily tours

participation rate

travel agency

tourists

non-participation