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