Create a three-way Venn Diagram using A, B, C for the first letter of each language. Add the values that we know, which are only A (A=20), only B (B=27), and only C (C=26). There are four areas here where they cross over, so there are four unknowns:
People taking exactly Arabic & Bulgarian (AB)
People taking exactly Bulgarian & Chinese (BC)
People taking exactly Arabic & Chinese (AC)
People taking all three (ABC)
Now, see how anyone who is taking Arabic can also be taking another language, or even all three. We can create three equations, one for each language:
20+AB+AC+ABC=32
27+AB+BC+ABC=40
26+BC+AC+ABC=36
Since we have four unknowns, we will need four equations to solve for any of the values. Luckily, they also tell us that there are 11 students taking Arabic & Bulgarian, but some of them also take Chinese. Thus we have:
4. AB+ABC=11
Start with that last equation since there are fewer variables. Pick one variable to solve for:
AB = 11-ABC
Plugging into equation 1:
20+11-ABC+AC+ABC=32
31+AC=32
AC=1
Plugging into equation 3:
26+BC+1+ABC=36
27+BC+ABC=36
BC=9-ABC
Plugging all we know into equation 2:
27+ 11-ABC+9-ABC+ABC=40
47-ABC=40
ABC=7
Use this to solve for AB & BC:
AB=4 & BC=2
So there are 7 students taking all languages. To find out how many are taking no languages, take 100 and subtract from it your A, B, C, AB, BC, AC & ABC because none of these values overlap students. You should get 13.
