# In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

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In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15, French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study

(i) French only

(ii) English only

(iii) Sanskrit only

(iv) English and Sanskrit but not French

(v) French and Sanskrit but not English

(vi) French and English but not Sanskrit

(vii) at least one of the three languages

(viii) none of the three languages

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Let F be the set of students who study French, E be the set of students who study English and S be the set of students who study Sanskrit.

Then,

n{U) = 50, n(F) =17, n{E) = 13, and n{S) = 15, n(F ∩ E) = 9, n(E ∩ S) = 4, n(F ∩ S) = 5, n(F ∩ E ∩ S) = 3  (i) Number of students studying French only = e = 6

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

(iii) Number of students studying 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 = 30

(viii) Number of students studying none of the three languages but not French = 50-30 = 20

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