Q:

Principle of Inclusion-Exclusion missing variables

A survey of 100 students taken over the last year revealed that

30 of them smoke
50 of them drink
20 of them smoke and drink
25 go to the gym regularly, and of these, none smokes nor drinks.

Calculate how many of the 100 students neither smoke, nor drink, nor go to the gym.

So by the Principle of Inclusion-Exclusion, we should u
$$A^{c}B^{c}C^{c} = |U|-(|A|+|B|+|C|)+(|AB|+|AC|+|BC|)-|ABC|$$
$$= 100 - (30+50+25)+(20+X+Y)-Z$$
This is where I get stuck...

A:

It ought to be 15. I worked this out using a Venn diagram, as follows:

Draw a rectangle and write 100 in one of its corners. That's the whole student body.
Draw two intersecting circles, one for smokers and one for drinkers. Write 20 in the overlap, for those who drink and smoke. Now, the remainder of the smoking circle is 10 people, so write that in; the remainder of the drinking circle is 30 people, so write that in.
Write 25 outside both circles for the people who go to the gym.

Now you can see that the total number of people contained in the circles is $10 + 20 + 30 = 60$. Then you have the 25 people at the gym. The total is 85, so 15 people do nothing.

