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 solution please
To solve this question, we can use the principle of inclusion-exclusion.
a. Find the number of students that do:
i. Two subjects only: To find the number of students who do exactly two subjects, we need to subtract the students who do all three subjects from the total number of students who do at least one subject.
Total number of students who do exactly two subjects = (students who do only Arts and Biology) + (students who do only Arts and Chemistry) + (students who do only Chemistry and Biology)
Total number of students who do exactly two subjects = 4 + 3 + 5 = 12
ii. Exactly one subject: To find the number of students who do exactly one subject, we need to subtract the students who do more than one subject from the total number of students who do at least one subject.
Total number of students who do exactly one subject = (students who do only Arts) + (students who do only Chemistry) + (students who do only Biology)
Total number of students who do exactly one subject = 24 - 12 + 22 - 12 + 20 - 12 = 30
iii. At least two of the subjects: To find the number of students who do at least two subjects, we need to find the total number of students who do all three subjects and the students who do exactly two subjects.
Total number of students who do at least two subjects = (students who do all three subjects) + (students who do exactly two subjects)
Total number of students who do at least two subjects = 3 + 12 = 15
b. Representing the information on a complete Venn diagram:
A complete Venn diagram represents the relationships between the subjects and the number of students who do each combination of subjects.
Let's denote Arts as A, Chemistry as C, and Biology as B.
B (Biology)
/ | \
/ | \
4 3 5
∪____∪___∪___∪
A | 4 | 3 | 3
∪____∪___∪___∪
C 2 ∪____∪ 5
C∩A
In the Venn diagram above, the overlapping areas represent the number of students who do multiple subjects, and the non-overlapping areas represent the number of students who do only one subject. The numbers in each region represent the number of students in that specific combination.
Hope this helps! Let me know if you have any further questions.