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

There are 17 students.

• 11 students like one pizza topping

• 7 students like two of the toppings

• 4 students like 3 of the toppings

• 2 students like 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

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Can you show us the calculation you did, even if it is wrong? – Adriano Jun 10 '13 at 23:24
Are you sure of the wording of the problem? The only way that I see to make any sense of those numbers is to assume that when it says that $n$ students like $k$ of the toppings, it means that there are $n$ students who like at least $k$ of the toppings. In that case there are $17-11=6$ who like none. – Brian M. Scott Jun 10 '13 at 23:28
Perhaps this is one of a set of questions asking how many people like exactly 0, 1, 2, 3, etc. toppings. – robjohn Jun 10 '13 at 23:43

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!.

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I hate problems that put in a bunch of distractions - nice sorting through those! +1 – Amzoti Jun 11 '13 at 2:36

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 $$17-11=6.$$

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