In a class of 40 students, 22 study art, 18 study biology and 6 study only chemistry. 5 study all 3 subjects, 9 study art and biology, 7 study art but not chemistry and 11 study exactly one subject. illustrate the info on a Venn diagram. how many students study exactly two subjects. how many students study none of the subjects. what is the probability that a student chosen at random study chemistry.

draw 3 intersecting circles, label them A, B, C, for the subject names
fill in from the centre outwards
you are told 5 take all three, so put 5 in the intersection of all 3
Then do the "two at a time" sections of the Venn diagram
e.g. 9 study art and chemistry. BUT, we already have 5 counted
in that part, so put 4 in the part representing only art and biology.
But 6 in the "only chemistry" part.

We don't know the art and chemistry, but not biology, call that x
Since we know art has 22, we know that the "only art" part would be
22 - 4-4-x = 13-x

We know that 7 study art but not chemistry
13-x + 4 = 7
x = 10

replace the x with 10

Label the "only chemistry and biology" part as y
from our data:
then the only biology part is
18-4-5-y = 9-y

we know 11 study exactly one subject
13-x + 6 + 9-y = 11
13-10 + 6 + 9 -y = 11
y = 7

replace all the y's with 7

looks like all the part are filled in, but I only count 37, and in the class of
40 students, 3 must not take any of the listed subjects.
counting up the chemistry numbers, there are 28

so the prob that a random student of the class selected takes chemistry
= 28/40
= 7/10 or 0.7

check my arithmetic, the numbers seem to make sense