Question

A class has 175 students. the followng data shows the number of students obtaining one or more subjects. mathematics 100; physics 70; chemistry 40; mathematics and physics 30; mathematics and chemistry 28; physics and chemistry 23; mathematics, physics and chemistry 18. how many students have offered mathematics alone?

(1) 35

(2) 48

(3) 60

(4) 22

Solution

Solution:-

Let M, P and C represents the sets of the students who studied mathematics, physics and chemistry respectively.

Given : n(M) = 100 ; n(P) = 70 and n(C) = 40

n(M∩P) = 30 ; n(M∩C) = 28 and n(P∩C) = 23 and n(M∩P∩C) = 18

n(M∪P∪C) = n(M) + n(P) + n(C) - n(M∩P) - n(M∩C) - n(P∩C) + n(M∩P∩C)

= 100 + 70 + 40 - 30 - 28 - 23 + 18

= 228 - 81

= 147

So, the number of students who any of the subjects = 147

Therefore, the number of students who have not any of these three subjects = 175 - 147

= 28 students.

Thus, 28 students have not offered any of these three subjects.

Now,

Number of students who are enrolled in mathematics alone = n(M) - n(M∩P) - n(M∩C) + n(M∩P∩C)

= 100 - 30 - 28 +18

= 118 - 58

= 60 students enrolled in mathematics alone.

Let M, P and C represents the sets of the students who studied mathematics, physics and chemistry respectively.

Given : n(M) = 100 ; n(P) = 70 and n(C) = 40

n(M∩P) = 30 ; n(M∩C) = 28 and n(P∩C) = 23 and n(M∩P∩C) = 18

n(M∪P∪C) = n(M) + n(P) + n(C) - n(M∩P) - n(M∩C) - n(P∩C) + n(M∩P∩C)

= 100 + 70 + 40 - 30 - 28 - 23 + 18

= 228 - 81

= 147

So, the number of students who any of the subjects = 147

Therefore, the number of students who have not any of these three subjects = 175 - 147

= 28 students.

Thus, 28 students have not offered any of these three subjects.

Now,

Number of students who are enrolled in mathematics alone = n(M) - n(M∩P) - n(M∩C) + n(M∩P∩C)

= 100 - 30 - 28 +18

= 118 - 58

= 60 students enrolled in mathematics alone.

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