Answers:
number who study exactly one subject = 62number who study at least one subject = 92number who study none of the subjects = 28
The venn diagram is shown below.
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Explanation:
M = set of people studying math
C = set of people studying chemistry
P = set of people studying physics
Something like the notation n(C) refers to the count of items in set C, aka the number of people who study chemistry. Similar ideas apply to n(M) and n(P) to refer to the counts of math and physics respectively.
Something like n(C and M) is the number of people who study chemistry and math. Same idea applies to something like n(C and P) or n(M and P)
With all that in mind, we have these 7 facts
n(M) = 41
n(C) = 48
n(P) = 42
n(C and M) = 16
n(M and P) = 14
n(C and P) = 18
n(C and M and P) = 9
We'll start with fact 7. This value goes in the very center of the 3 overlapping circles of the venn diagram. Refer to the diagram below.
Since 9 people study all three subjects, and 16 study chemistry and math, this must mean 16-9 = 7 people study chemistry and math without physics involved. We'll write 7 in the region between circles C and M, but outside of circle P.
Since 9 people study all three subjects, and 14 study math and physics, we know that 14-9 = 5 people study only math and physics without chemistry. We'll write 5 in the region between circles M and P, but outside circle C.
Since 9 people study all three subjects, and 18 study chemistry and physics, we know that 18-9 = 9 people study only chemistry and physics without math. We'll write 9 in the region between circles C and P but outside circle M.
If you're lost, then refer the diagram below. So far, we have filled in the very center "9" and the closest three values surrounding it (the 7, 9 and 5).
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Focus your attention on circle M for now. The values in this circle add to: 7+9+5 = 21
Since n(M) = 41, this must mean the missing value must be 20. This goes in circle M but outside the other two circles.
Note how 7+9+5+20 = 41 when we add everything in circle M. This circle is completely filled out. We'll follow a similar path for the other circles.
Move onto circle C. Adding everything in it so far gets us 9+9+7 = 25. We want to get to n(C) = 48 which means we need to have 48-25 = 23 in circle C but outside the other circles. Circle C is now done.
Lastly, focus on circle P. The values we found so far in this circle add to 9+9+5 = 23. The missing value must be 19 because 42-23 = 19, and n(P) = 42.
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At this point, all three circles are filled out. The only thing missing is the number outside the circles, but inside the rectangle.
Add up everything in the circles: 23+7+20+9+9+5+19 = 92
This tells us that we have 92 people who studied at least one subject (i.e. one or more subjects). There are 120 people surveyed, meaning that 120-92 = 28 people study neither of these subjects. We write 28 outside the circles.
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The venn diagram is done by this point. To address the questions, we'll just read from the diagram (and at most do an addition problem).
To find the number of people who study exactly one subject, we will add up the numbers that are in one circle but not in any other circle. Those values are: 23, 20, and 19. So 23+20+19 = 62 people study exactly one subject.
We already found the number of people who study at least one subject, and that was 92 people. This was when we added every number in the circles.
For the last question, we already found this value as well. This answer is 28 people which is found outside all three circles.
The venn diagram is shown below to visually summarize and organize all of the data. Hopefully it clears up any confusion you may have about the previous steps.
