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
With explanation please
i. The number of students that do two subjects only is 8 (4 do Arts and Biology only, 3 do Arts and Chemistry only, and 5 do Chemistry and Biology only).
ii. The number of students that do exactly one subject is 22 (24 do Arts, 22 do Chemistry, and 20 do Biology).
iii. The number of students that do at least two of the subjects is 28 (3 do all three subjects, 4 do Arts and Biology only, 3 do Arts and Chemistry only, and 5 do Chemistry and Biology only).
To find the number of students that do two subjects only, exactly one subject, and at least two of the subjects, we can use the principle of inclusion-exclusion.
a. Find the number of students that do two subjects only:
First, we add up the numbers of students who do Arts and Biology only, Arts and Chemistry only, and Chemistry and Biology only:
4 + 3 + 5 = 12 students.
b. Find the number of students that do exactly one subject:
To calculate this, we need to determine the number of students who do each subject individually:
Number of students doing Arts only = (total number of students doing Arts) - (number of students doing Arts and Biology only) - (number of students doing Arts and Chemistry only) + (number of students doing all three subjects)
= 24 - 4 - 3 + 3 = 20 students.
Similarly, we calculate the number of students doing Chemistry only and Biology only:
Number of students doing Chemistry only = (total number of students doing Chemistry) - (number of students doing Chemistry and Biology only) - (number of students doing Arts and Chemistry only) + (number of students doing all three subjects)
= 22 - 5 - 3 + 3 = 17 students.
Number of students doing Biology only = (total number of students doing Biology) - (number of students doing Arts and Biology only) - (number of students doing Chemistry and Biology only) + (number of students doing all three subjects)
= 20 - 4 - 5 + 3 = 14 students.
To find the total number of students doing exactly one subject, we sum up the number of students doing each subject individually:
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
= 20 + 17 + 14 = 51 students.
c. Find the number of students that do at least two of the subjects:
To find this, we need to sum up the numbers of students who do two subjects only and the number of students doing all three subjects:
Number of students doing at least two subjects = Number of students doing two subjects only + Number of students doing all three subjects
= 12 + 3 = 15 students.
To summarize:
i. The number of students that do two subjects only is 12.
ii. The number of students that do exactly one subject is 51.
iii. The number of students that do at least two of the subjects is 15.