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Hello, this might be a long question, but I think it is quite easy to do and I haven done it already,
but apparently I got my answer wrong (16), can someone please explain to me why the answer is 12?
Knowledge I have to know to solve this question:
the inclusion-exclusion principle for three sets and Venn Diagrams
Question:
A group of 82 senior students all study at least one of Biology, Chemistry and Physics. The number of students
who study all three subjects is equal to the number who study Biology and Chemistry but not Physics.
The number of students who study all three subjects is also equal to the number who study Chemistry and Physics
but not Biology. Finally, there are twice as many students who study Biology and Physics but not Chemistry,
as those who study all three subjects.
If the numbers of students studying Biology, Physics and Chemistry are 41, 35, and 30 respectively,
determine the number of students who study both Biology and Physics.
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Draw (= sketch) Venn diagram. I assume you know what is it, and I should not explain the basics to you.
The diagram has three circes : B, C and P, they are your primary sets.
You are given |B| = 41, |P| = 35, |C| = 30.
Let x be the number of elements in triple intersection BCP: x = |BCP|.
Then from the condition
x = BC \ BCP
x = CP \ BCP
2x = BP \ BCP
Here the symbols of two and three letters mean INTERSECTIONS of corresponding subsets; the symbol " \ " means subtraction of sets (subsets).
Mark everything what I said, in the diagram.
From what I said above, you have
|BC| = 2x,
|BP| = 3x,
|CP| = 2x.
Now I write inclusion-exclusion principle / (equation)
|B U C U P| = B + C + P - BC - BP - CP + BCP,
or
82 = 41 + 35 + 30 - 2x - 3x - 2x + x.
It gives
2x + 3x + 2x - x = 41 + 35 + 30 - 82, or 6x = 24, which implies x = 4.
So, the triple intersection BCP contains 4 elements: BCP = 4.
The intersection BP, which is under the problem's question, contains 2x + x = 3x elements = 3*4 = 12. ANSWER
Solved and explained.
If you need more explanations or if you want to see MANY (tens) SIMILAR (or different) solved problems
at this site - then let me know, I will share the links with you.