How many students like none of the toppings? (Principle of Inclusion - Exclusion)

Question

There are 17 students.

• 11 students like at least one pizza topping

• 7 students like at least two of the toppings

• 4 students like at least 3 of the toppings

• 2 students like at least 4 of the toppings

• 1 students likes all of the toppings

How many students like none of the toppings?

I tried adding up all the sets then subtracting the overlaps then adding back in the intersections but I miscounted somewhere because I got way more than 17 which isn't possible because there are only 17 students

Answer 1

Most of the numbers given are distractions (if the problem is only interested in the number of people who don't like any of the toppings). The thing to note, here, is that the numbers cannot be describing distinct sets of people, since we've only got $17$--in particular, $11$ people like at least one of the toppings, $7$ like at least $2$ of the toppings, and so on.

The people who don't like any of the toppings are the people who don't like at least one of the toppings. The only numbers that matter here are $17$ and $11$, which give the answer by $$17-11=6.$$

Answer 2

Given the wording of the problem, you can take the one and only person liking all toppings to also like 4 toppings, and 3 toppings...and 2 and 1.

Likewise, those liking $4$ toppings also like 3 toppings, ... 2, and 1. Etc.

Notice that

"those who like 1 topping" $\supset$ those who like 2 $\supset$ those who like 3 toppings $\supset$ and so on.

The ONLY numbers that matter in answering the question are those liking NO topping, so we exclude 11, since 11 students like at least one topping (and it happens to be the case that of those 11 students, some like more toppings, too). Given there are $17$ students, excluding the $11$ topping-loving students from $17$ gives us $17 - 11 = 6$ students who must not like any extraneous toppings!.