Question

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

French $=17$, English $=13$, Sanskrit $=15$

French and English $=9$, 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 no English

(vi) French and English but no Sanskrit

(vii) at least one of the three languages

(viii) none of the three language

Answer 1

Lets take the languages of,

French $=$ set $A$,

English $=$ set $B$,

Sanskrit $=$ set $C$

$n\left(A\right)=17,n\left(B\right)=13,n\left(C\right)=15$

$n(A\cap B)=9$, $n(B\cap C)=4,n(A\cap C)=5$

$n(A\cap B\cap C)=3$

$n\left(u\right)=50$

$n(A\cap \overline{B}\cap \overline{C})=n\left(A\right)-n(A\cap B)-n(A\cap C)+n(A\cap B\cap C)$

$=17-9-5+3$

French only $=\mathrm{6...........}$ (Answer i)

$n(\overline{A}\cap B\cap \overline{C})=n\left(B\right)-n(B\cap C)-n(A\cap B)+n(A\cap B\cap C)$

$=13-4-9+3$

English only $=3$...........(Answer ii)

$n(\overline{A}\cap \overline{B}\cap C)=n\left(C\right)-n(A\cap C)-n(B\cap C)+n(A\cap B\cap C)$

$=15-4-5+3$

Sanskrit only $=9$........... (Answer iii)

$n(\overline{A}\cap B\cap C)=n(B\cap C)-n(A\cap B\cap C)$

French, English and Sanskrit $=4-3=1$...........

$n(A\cap \overline{B}\cap C)=n(A\cap C)-n(A\cap B\cap C)$

$=5-3=2$........... (Answer v)

$n(A\cap B\cap \overline{C})=n(A\cap B)-n(A\cap B\cap C)$

$=9-3=6$........... (Answer vi)

$n(A\text{or}B\text{or}C$ ) =$5a-(A\cap B\cap C)$

$=50-\left\{n\right(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C\left)\right\}$

$=50-\{17+13+15-5-4-5+3\}$

$=20$........... (Answer vii)

$\therefore n$(at least of $A$ or $B$ or $C$)=$50-20=30$........... (Answer viii)