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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

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

Answers (1)

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
n(U) = 5, n(F) = 17, n(E) = 13, n (S) = 15, n(F\cap E) = 9, n(E\cap S) = 5 and n(F\cap E\cap S) = 3

\begin{aligned} n(F\cap E\cap S) = 3&\Rightarrow e = 3\\ n(F\cap E) = 9&\Rightarrow b + e = 9&\Rightarrow b = 6 \\ n(F\cap S) = 5 &\Rightarrow d + e = 5 &\Rightarrow d = 2 \\ n(F) = 17 &\Rightarrow a + b +d + e = 17&\Rightarrow a = 6\\n(E) = 13&\Rightarrow b + c + e +f = 13&\Rightarrow c = 3\\ n(S) = 15 &\Rightarrow d + e + f + g = 15&\Rightarrow g = 9\end{aligned}

Number of students who study French only =a =6
Number of students who study English only =c =3 
Number of students who study Sanskrit only =g =9
Number of students who study English and Sanskrit but not French =f =1
Number of students who study French and Sanskrit but not English =d =2
Number of students who study French and English but not Sanskrit =b = 6
Number of students who study at least one of the three languages = a+b+c+d+e+f+g =6+6+3+2+3+1+9 =30
Number of students who study none of the three languages = 50-30=20
 

 

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