A large circle represents all 200 students. Inside the circle are three smaller interlocking circles, each circle represents as subject, and is labelled with H (history), A (art) or S (science). The intersection of the smaller circles with each other produces 7 distinct regions. The region enclosed by the large circle but outside the others makes an eighth region. The regions are represented by a to h below and each letter stands for the number of students in the region.
a. A only students
b. H only
c. S only
d. A and H only
e. H and S only
f. A and S only
g. All 3
h. None
We combine the letters according to the facts given. A diagram helps. The combination of letters below means the corresponding regions' numbers of students are added together. So we have:
H: bdeg=70; S: cefg=80; A: adfg=60; A+H: dg=35; S+A: fg=31; All: g=15.
From these we find:
f=16 (fg-g=31-15); d=20; ad=29 (adfg-fg) so a=9; be=35; ce=49; abcdefgh=200, ch=200-(adfgbe)=200-(60+35)=105.
h=105-c=105-49+e=56+e.
There is insufficient data to find h, the number of students having neither art, history or science, because we need to know c, e, or b (science only, science and history, or history only). e must be between 0 and 35. If there are no students taking science and history, but not art, then there are 56 students taking none of the subjects. If no students take only history, then e=35 and 91 students take none of the subjects.