Question

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 an Sanskirt $=3$.
Find the number of students who study

$\left(i\right)$ French only

$\left(ii\right)$ English only

$\left(iii\right)$ Sanskrit only

$\left(iv\right)$ English and Sanskrit but not French

$\left(v\right)$ French and Sanskrit but not English

$\left(vi\right)$ French and English but not Sanskrit

$\left(vii\right)$ at least one of the three languages

$\left(viii\right)$ none of the three languages

Answer 1

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

Given : $n\left(U\right)=50,n\left(F\right)=17,n\left(E\right)=13$,
$n\left(S\right)=15,n(F\cap E)=9,n(E\cap S)=4,$
$n(F\cap S)=5,n(F\cap E\cap S)=3$

Drawing the Venn diagram


From the figure, we have

$a=n(E\cap F\cap S)=3,$

$a+d=n(F\cap S)=5$

$\Rightarrow d=2$

$a+b=n(F\cap E)=9$

$\Rightarrow b=6$

$a+c=n(S\cap E)=4$

$\Rightarrow c=1$

$a+b+d+e=n\left(F\right)=17$

$\Rightarrow 3+6+2+e=17$

$\Rightarrow e=6$

$a+b+c+g=n\left(E\right)=13$

$\Rightarrow 3+6+1+g=13$

$\Rightarrow g=3$

$a+c+d+f=n\left(S\right)=15$

$\Rightarrow 3+1+2+f=15$

$\Rightarrow f=9$

$\therefore a=3,b=6,c=1,d=2e=6,$

$f=9,g=3$

Number of students studying French only $=e=6$

Number of students studying English only $=g=3$

Number of students studying Sanskrit only $=f=9$

Number of students studying English and Sanskrit but not French $=c=1$

Number of students studying French and Sanskrit but not English $=d=2$

Number of students studying English and French but not Sanskrit $=b=6$

Number of students studying at least one of the three languages

$=a+b+c+d+e+f+g=30$

Number of students studying none of the three languages $=50-30=20$