Answer :

**Given:**

Total number of students = 100

Number of students passed in English = 15

Number of students passed in Mathematics = 12

Number of students passed in Science = 8

Number of students passed in English and Mathematics = 6

Number of students passed in Mathematics and Science = 7

Number of students passed in English and Science = 4

Number of students passed in all three = 4

Let U be the total number of students, E, M and S be the number of students passed in English, Mathematics and Science respectively

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

(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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