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Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science; 4 in English and Science; 4 in all the three. Find how many passed

(i) in English and Mathematics but not in Science

(ii) in Mathematics and Science but not in English

(iii) in Mathematics only

(iv) in more than one subject only

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Best answer

According to the question,

Total number of students = 100

Number of students who passed in English = 15

Number of students who passed in Mathematics = 12

Number of students who passed in Science = 8

Number of students who passed in English and Mathematics = 6

Number of students who passed in Mathematics and Science = 7

Number of students who passed in English and Science = 4

Number of students who passed in all three = 4

Let the total number of students = U

Let the number of students passed in English = E

Let the number of students passed in Mathematics = M

Let the number of students passed in Science = S

n(M ∩ S ∩ E) = a = 4

n(M ∩ S) = a + d = 7

⇒ 4 + d = 7

⇒ d = 3

n(M ∩ E) = a + b = 6

⇒ 4 + b = 6

⇒ b = 2

n(S ∩ E) = a + c = 4

⇒ 4 + c = 4

⇒ c = 0

n(M) = e + d + a + b = 12

⇒ e + 4 + 3 + 2 = 12

⇒ e + 9 = 12

⇒ e = 3

n(E) = g + c + a + b = 15

⇒ g + 0 + 4 + 2 = 15

⇒ g + 6 = 15

⇒ g = 9

n(S) = f + c + a + d = 8

⇒ f + 0 + 4 + 3 = 8

⇒ f + 7 = 8

⇒ f = 1

Therefore, from the above equations, we get that,

(i) Number of students passed in English and Mathematics but not in Science = b = 2

(ii) Number of students in Mathematics and Science but not in English = d = 3

(iii) Number of students in Mathematics only = e = 3

(iv) Number of students in more than one subject only = a + b + c + d = 4 + 3 + 2 + 0 = 9

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