Discrete Mathematics problem concerning sets

Question

A class of 25 students has options as to which science subjects they study. The options are biology, chemistry, geology and physics. 15 of the 25 students took only two of these sciences, whilst the other 10 students took three. 5 students studied geology and 8 studied biology, but nobody took both. How many students studied chemistry, given that they all also took physics? Use B, C, G and P to denote the sets of students who studied biology, chemistry, geology and physics. You MUST answer this question by using the principle of inclusion-exclusion (as well as other deductions).

Answer 1

10 students took 3 subjects.

n(BG) = 0, so n(CP and 3 subjects) = 10

n(G) = 5, n(B) = 8, and 10 of these studied CP, and everyone took at least 2 subjects, so n(B U G and 2 subjects) = 5 + 8 - 10 = 3

Then, as n(2 subjects) = 15, and there are only 4 subjects available, the only 2 subjects available not including B and G are C and P, so 15 - 3= 12 studied CP and 2 subjects

Then, the total number studying C and P is 12 + 10 = 22.