2. In a class of 48 students, 24 of them do Arts, 22 do Chemistry and 20 do Biology. All the students do at least one of the three subjects. 3 do all three subjects while 4 do Arts and Biology only, 3 do Arts and Chemistry only and 5 do Chemistry and Biology only.
a. Find the number of numbers of students that do
i. two subjects only
ii. exactly one subject
iii. at least two of the subjects
b. Represent the information on a complete Venn diagram.
With explanation and solution please
a.
i. To find the number of students that do two subjects only, we need to add up the number of students that do Arts and Biology only, Arts and Chemistry only, and Chemistry and Biology only.
4 students do Arts and Biology only.
3 students do Arts and Chemistry only.
5 students do Chemistry and Biology only.
Therefore, the number of students that do two subjects only is 4 + 3 + 5 = 12.
ii. To find the number of students that do exactly one subject, we need to subtract the students that do two subjects only and the students that do all three subjects from the total number of students.
There are 3 students that do all three subjects.
The number of students that do exactly one subject is 48 - 12 - 3 = 33.
iii. To find the number of students that do at least two of the subjects, we need to add the students that do all three subjects and the students that do two subjects only.
There are 3 students that do all three subjects, and 12 students that do two subjects only.
The number of students that do at least two of the subjects is 3 + 12 = 15.
b. Here is a representation of the information on a complete Venn diagram:
```
/\ _________
Arts | \ | |
3|___\___________|__Biology|
| \_________|__ ___|
| ___|_____|_____
| | Chemistry |
| | ________|
|__________|______|
```
Explanation: The Venn diagram shows the three subjects - Arts, Biology, and Chemistry. The overlapping regions represent the students who do multiple subjects. The number at the intersection of two subjects represents the number of students who do both of those subjects. For example, at the intersection of Arts and Biology, there are 4 students who do Arts and Biology only. At the intersection of Biology and Chemistry, there are 5 students who do Biology and Chemistry only. The number at the center represents the number of students who do all three subjects, which is 3 in this case.
To find the number of students that do different combinations of subjects, we can use the formula for the number of elements in the union of three sets:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)
Given information:
n(A) = 24
n(B) = 22
n(C) = 20
n(A ∩ B ∩ C) = 3
n(A ∩ B) = 4
n(A ∩ C) = 3
n(B ∩ C) = 5
a. Determining the number of students for different subject combinations:
i. Two subjects only:
n(A ∩ B) + n(A ∩ C) + n(B ∩ C) = 4 + 3 + 5 = 12
ii. Exactly one subject:
n(A) + n(B) + n(C) - 2 * (n(A ∩ B) + n(A ∩ C) + n(B ∩ C)) - n(A ∩ B ∩ C) = 24 + 22 + 20 - 2 * (4 + 3 + 5) - 3 = 48 - 24 - 3 = 21
iii. At least two of the subjects:
n(A ∩ B ∩ C) = 3
b. Representing the information on a complete Venn diagram:
The Venn diagram can be drawn as follows:
____A____
| |
_____________
| | | |
B AB C BC
| | | |
‾‾‾‾‾‾‾‾‾‾‾‾
| |
‾‾‾‾‾‾‾‾
Here,
A represents the number of students doing Arts (24)
B represents the number of students doing Biology (20)
C represents the number of students doing Chemistry (22)
AB represents the number of students doing Arts and Biology only (4)
BC represents the number of students doing Biology and Chemistry only (5)
AC represents the number of students doing Arts and Chemistry only (3)
A ∩ B ∩ C represents the number of students doing all three subjects (3)
The numbers outside the circles (12) represent the number of students doing two subjects only (combination of AB, AC, and BC).
The numbers inside the circles (21) represent the number of students doing exactly one subject (A + B + C - 2 * (AB + AC + BC) - (A ∩ B ∩ C)).
The number of students left (A + B + C) - (AB + AC + BC + (A ∩ B ∩ C) + 2 * (AB + AC + BC)) = 48 - 21 - 3 - 2 * 12 = 48 - 21 - 3 - 24 = 48 - 48 = 0, represents the number of students who do all three subjects.
Note: The numbers in the diagram may be adjusted based on the available space and clarity.