Answer :

**Given:**

Total number of students = 50

Number of students studying English = 13

Number of students studying French = 17

Number of students studying Sanskrit = 15

Number of students studying English and French = 9

Number of students studying French and Sanskrit = 5

Number of students studying English and Sanskrit = 4

Number of students studying all three subjects = 3

Let U be the total number of students, E, F and S be the number of students passed in English, French and Sanskrit respectively

n(F ∩ S ∩ E) = a = 3

n(F ∩ S) = a + d = 5

⇒ 3 + d = 5

⇒ d = 2

n(F ∩ E) = a + b = 9

⇒ 3 + b = 9

⇒ b = 6

n(S ∩ E) = a + c = 4

⇒ 3 + c = 4

⇒ c = 1

n(F) = e + d + a + b = 17

⇒ e + 2 + 3 + 6 = 17

⇒ e + 11 = 17

⇒ e = 6

n(E) = g + c + a + b = 13

⇒ g + 1 + 3 + 6 = 13

⇒ g + 10 = 13

⇒ g = 3

n(S) = f + c + a + d = 15

⇒ f + 1 + 3 + 2 = 15

⇒ f + 6 = 15

⇒ f = 9

(i) Number of students study French only = e = 6

(ii) Number of students study English only = g = 3

(iii) Number of students study Sanskrit only = f = 9

(iv) Number of students studying English and Sanskrit but not French = c = 1

(v) Number of students studying French and Sanskrit but not English = d = 2

(vi) Number of students studying French and English but not Sanskrit = b = 6

(vii) Number of students studying at least one of the three languages

= a + b + c + d + e + f + g

= 3 + 6 + 1 + 2 + 6 + 9 + 3

= 30

(viii) Number of students studying none of the three languages = Total – (a + b + c + d + e + f + g)

= 50 – (3 + 6 + 1 + 2 + 6 + 9 + 3)

= 50 – 30

= 20

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