Part (a)
B = biology
C = chemistry
P = physics
n(B) = number of people in biology
n(C) = number of people in chemistry
n(P) = number of people in physics
Given facts:
- Fact 1: 130 study Biology
- Fact 2: 135 study Chemistry
- Fact 3: 115 study Physics
- Fact 4: 86 study Biology and Chemistry
- Fact 5: 70 study Chemistry and Physics
- Fact 6: 64 study Physics and Biology
n(C) = 135 from Fact 2
n(P) = 115 from Fact 3
n(B and C) = 86 from Fact 4
n(C and P) = 70 from Fact 5
n(B and P) = 64 from Fact 6
There are 200 students total, and they take at least one of the courses mentioned.
This means there aren't any students who don't take one of the courses.
n(B or C or P) = 200
Use the inclusion-exclusion principle
n(B or C or P) = n(B)+n(C)+n(P) - n(B and C) - n(C and P) - n(B and P) + n(B and C and P)
200 = 130+135+115 - 86 - 70 - 64 + n(B and C and P)
200 = 160 + n(B and C and P)
n(B and C and P) = 200 - 160
n(B and C and P) = 40
Answer: There are 40 students who take all 3 subjects.
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Part (b)
We'll use the result of part (a) to write the following:
n(B and C, not P) = n(B and C) - n(B and C and P)
n(B and C, not P) = 86 - 40
n(B and C, not P) = 46
There are 46 students in biology and chemistry, but not in physics.
and,
n(C and P, not B) = n(C and P) - n(B and C and P)
n(C and P, not B) = 70 - 40
n(C and P, not B) = 30
There are 30 students in chemistry and physics, but not in biology.
and,
n(B and P, not C) = n(B and P) - n(B and C and P)
n(B and P, not C) = 64 - 40
n(B and P, not C) = 24
There are 24 students in biology and physics, but not in chemistry.
Adding the three results gets us
46+30+24 = 100
Answer: There are 100 students in exactly two subjects.
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Part (c)
Use the inclusion-exclusion principle.
n(B only) = n(B) - n(B and C) - n(B and P) + n(B and C and P)
n(B only) = 130 - 86 - 64 + 40
n(B only) = 20
Answer: 20 students study only biology
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The venn diagram is shown below
Take note how the numbers are placed in the 8 regions.
For example, we have "46" in the region inside B and C, but outside P to represent the 46 students taking bio and chem, but not physics.
Also notice that
- The numbers in the B circle add to 130
- The numbers in the C circle add to 135
- The numbers in the P circle add to 115
- All of the numbers add to the grand total of 200
We have 0 outside all the circles to represent the idea that everyone is taking at least one of the three mentioned courses.
The venn diagram may also help show how/why the inclusion-exclusion principle works.
The "exclusion" part is us subtracting off something like n(B and C) and then the "inclusion" part is to counterbalance things by adding in something like n(B and C and P).