# A survey of 100 students produced the following statics: 32 study mathematics, 20 study physics, 45 study Biology, 15 study mathematics and biology, 7 study mathematics and physics, 10 study physics and Biology, 30 do not study any of the three subjects

1, Present the information in a Venn Diagram
ii, Find the number of students who study all the three subjects
iii, Take exactly one of the three subjects

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1. We don't know the intersection of all three circles, so label it x
Now go to the intersection of two of the circles, say, the M and B circles
It says that sum is 15, but we have already counted x of them, so in the remaining part of the football-shaped region is 15-x
Do that for the other two intersection of pairs to have 10-x and 7-x

Now go to the whole circles, start with M
We have already counted, 15-x, x, and 7-x
or 22-x
But the whole math circle contains 32, so the open part must be 32-(22-x) = 10+x
Do the same with the other whole circles and get
20+x and 3+x

Now add them all up.
We can use a short-cut here, we know that everything in the Math circle is 32, so ..

32 + 10-x + 20+x + 3+x = 70
x = 5

ii) 5 take all three subjects
ii) add up all the parts in the outer parts of the circle which do not intersect with any other circles
that is,
Math only = 27
Biology only = 25
Physics only = 8

or

using the formula:
N(M or P or B) = N(M) + N(B) + N(P) - N(M and P) - N(M and B) - N(P and B) + N(M and P and B)
70 = 32+45+20 - 7 - 15 -10 + x
x= 5

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2. 屌你

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3. Maths only is 25 because 45+5-25
Bio only is 15 because 32+5-22
Physics only is 8 because 20+5-17

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