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

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

屌你

Maths only is 25 because 45+5-25

Bio only is 15 because 32+5-22
Physics only is 8 because 20+5-17

i) Here's a Venn Diagram representing the given information:

_______________________
| Mathematics |
| /|
| / |
| __________________ |
| | / | |
| | Biology / | |
| |_______________/ | |
| | | |
| | | |
| | | |
|_________|_______|____|
Physics

ii) To find the number of students who study all three subjects, we look at the overlapping region in the Venn Diagram where all three subjects intersect. From the given information, we know that 15 students study mathematics and biology, and 7 students study mathematics and physics. However, we don't have direct information about the number of students who study physics and biology.

So, the number of students who study all three subjects is unknown (unless the information about physics and biology is provided).

iii) To find the number of students who take exactly one of the three subjects, we need to subtract the students who study multiple subjects from the total number of students in each subject.

Mathematics only: 32 - 15 (Mathematics and Biology) - 7 (Mathematics and Physics) = 10

Physics only: 20 - 7 (Mathematics and Physics) - Unknown (Physics and Biology) = Unknown

Biology only: 45 - 15 (Mathematics and Biology) - Unknown (Physics and Biology) = Unknown

In this case, we are missing the necessary information to accurately determine the number of students who take exactly one of the three subjects.

To answer these questions, we will start by constructing a Venn diagram.

i) Venn Diagram:

A Venn diagram is a way to represent relationships between different sets of items. In this case, the three sets are Mathematics, Physics, and Biology.

Start by drawing three overlapping circles to represent these three subjects. Label the circles as Mathematics (M), Physics (P), and Biology (B).

Inside the circles, write the number of students who study each subject, along with any overlapping areas.

Based on the given information:
- 32 students study Mathematics (M)
- 20 students study Physics (P)
- 45 students study Biology (B)
- 15 students study both Mathematics and Biology (M ∩ B)
- 7 students study both Mathematics and Physics (M ∩ P)
- 10 students study both Physics and Biology (P ∩ B)
- 30 students do not study any of the three subjects (No overlapping region)

Now, let's move on to answering the specific questions:

ii) The number of students who study all three subjects (M, P, and B):
Looking at the Venn diagram, we can see that there is no specific data given for the students who study all three subjects. However, we can deduce this information by using the given data.

Start with the total number of students:
Total students = (M) + (P) + (B) - (M ∩ B) - (M ∩ P) - (P ∩ B) + (Not studying any subject)

Based on the given data, we can calculate:
Total students = 32 + 20 + 45 - 15 - 7 - 10 + 30
Total students = 95

Now, to find the number of students who study all three subjects:
Number of students who study all three subjects = Total students - (Not studying any subject)
Number of students who study all three subjects = 95 - 30
Number of students who study all three subjects = 65

Therefore, 65 students study all three subjects.

iii) Taking exactly one of the three subjects:
To find the number of students who take exactly one of the three subjects, we need to exclude those who study more than one subject.

Number of students taking exactly one subject = (M) + (P) + (B) - 2 × [(M ∩ B) + (M ∩ P) + (P ∩ B)] - (Not studying any subject)

Using the data provided, we can calculate:
Number of students taking exactly one subject = 32 + 20 + 45 - 2 × (15 + 7 + 10) - 30
Number of students taking exactly one subject = 32 + 20 + 45 - 2 × 32 - 30
Number of students taking exactly one subject = 45 - 64 - 30
Number of students taking exactly one subject = -49

The result is negative, which indicates that there might be an error in the given data or calculations. Please check the information provided and try again.