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.
Do the Venn diagram first
arts only = 24 - 3 - 3 - 4 = 14
chem only 22 - 3 - 3 - 5 = 11
bio only 20 -4 -3 -5 = 8
etc
To solve this problem, we can use the principle of inclusion-exclusion.
Given:
Total number of students = 48
Number of students doing Arts = 24
Number of students doing Chemistry = 22
Number of students doing Biology = 20
Number of students doing all three subjects = 3
Number of students doing Arts and Biology only = 4
Number of students doing Arts and Chemistry only = 3
Number of students doing Chemistry and Biology only = 5
a. Find the number of students that do:
i. Two subjects only:
To find the number of students doing exactly two subjects, we need to subtract those who are doing all three subjects.
Number of students doing exactly two subjects = (Number of students doing Arts and Chemistry only) + (Number of students doing Arts and Biology only) + (Number of students doing Chemistry and Biology only) - (Number of students doing all three subjects)
= 3 + 4 + 5 - 3
= 9
ii. Exactly one subject:
To find the number of students doing exactly one subject, we need to subtract those who are doing more than one subject from the total number of students.
Number of students doing exactly one subject = (Number of students doing Arts only) + (Number of students doing Chemistry only) + (Number of students doing Biology only) - (Number of students doing exactly two subjects) - (Number of students doing all three subjects)
= 24 + 22 + 20 - 9 - 3
= 54 - 12
= 42
iii. At least two of the subjects:
To find the number of students doing at least two subjects, we need to subtract those who are doing only one subject from the total number of students.
Number of students doing at least two subjects = Total number of students - Number of students doing exactly one subject
= 48 - 42
= 6
b. To represent the information on a complete Venn diagram, we need to draw three circles: one for Arts, one for Chemistry, and one for Biology. The circles should overlap in the following way:
Circle A: Represents the number of students doing Arts
Circle B: Represents the number of students doing Chemistry
Circle C: Represents the number of students doing Biology
The region where all three circles intersect represents the number of students doing all three subjects, which is 3.
The regions where pairs of circles intersect represent the number of students doing exactly two subjects:
- Circle A and Circle B intersection: Represents the number of students doing Arts and Chemistry only (3).
- Circle A and Circle C intersection: Represents the number of students doing Arts and Biology only (4).
- Circle B and Circle C intersection: Represents the number of students doing Chemistry and Biology only (5).
The regions outside the intersections represent the number of students doing only one subject:
- Outside all three circles: Represents the number of students doing none of the subjects.
- Outside Circle A: Represents the number of students doing only Chemistry and Biology.
- Outside Circle B: Represents the number of students doing only Arts and Biology.
- Outside Circle C: Represents the number of students doing only Arts and Chemistry.
I hope this helps! Let me know if you have any further questions.
To solve this problem, we can use the principle of Inclusion-Exclusion.
a. First, let's find the number of students who do two subjects only.
i. To find the number of students who do two subjects only, we can add up the number of students who do each combination of two subjects and subtract the number of students who do all three subjects.
For Arts and Chemistry only: 3
For Arts and Biology only: 4
For Chemistry and Biology only: 5
So, the number of students who do two subjects only is 3 + 4 + 5 = 12.
ii. To find the number of students who do exactly one subject, we need to find the number of students who do each subject individually and subtract the number of students who do two subjects only.
For Arts only: 24 - 3 - 4 = 17
For Chemistry only: 22 - 3 - 5 = 14
For Biology only: 20 - 4 - 5 = 11
So, the number of students who do exactly one subject is 17 + 14 + 11 = 42.
iii. To find the number of students who do at least two subjects, we can add up the number of students who do two subjects only and the number of students who do all three subjects.
Number of students who do two subjects only: 12
Number of students who do all three subjects: 3
So, the number of students who do at least two subjects is 12 + 3 = 15.
b. Now let's represent the information on a complete Venn diagram:
- Start by drawing a rectangle to represent the total number of students (48).
- Inside the rectangle, draw three overlapping circles to represent Arts, Chemistry, and Biology.
- Label the overlapping regions according to the given information.
- Write "3" in the region where all three subjects intersect.
- Write "4" in the region where Arts and Biology intersect but not Chemistry.
- Write "3" in the region where Arts and Chemistry intersect but not Biology.
- Write "5" in the region where Chemistry and Biology intersect but not Arts.
- Write the numbers we calculated earlier in the regions outside the overlaps:
- Write "17" in the region labeled "Arts only."
- Write "14" in the region labeled "Chemistry only."
- Write "11" in the region labeled "Biology only."
- Write "12" in the region labeled "Arts and Chemistry only."
- Write "12" in the region labeled "Arts and Biology only."
- Write "15" in the region labeled "Chemistry and Biology only."
This completes the Venn diagram representation of the given information.