Thread: simple set example

I can't figure out why the correct anwer is a)

Each of 35 students learns at least one of these languages: English, German, French.
5 students study all of these languages.
20 students study English.
15 students study German
10 students study French.


Decide which of these statement is true:


a) No student exists, who learns exactly 2 languages.
b) At least one student learns exactly 2 languages.
c) At least one student learns French and German and doesn't learn English.
d) Described situation cannot happen.
d) None of the statements is correct.


I would greatly appreciate if you describe how you got the result.
Thank you.

Let x, y and z denote (in some order) the number of students who study exactly two languages. Then by the inclusion–exclusion principle, we have

20 + 15 + 10 - (x + 5) - (y + 5) - (z + 5) + 5 = 35

from where x + y + z = 0.