Question

In a class of $60$ students $31$ students knows English, $23$ students knows French, $20$ students knows German,$8$ students knows all the three languages,$18$ students knows English and French, $13$ students knows English and German and $11$ knows French and German then the total number of students who doesn't knows any of the language and students who knows English and German but not French are

Answer 1

Let the students who knows English be $E$
Students who knows French be $F$
​​and the students who knows German be $G$
Then
$n(E\cup F\cup G)=n\left(E\right)+n\left(F\right)+n\left(G\right)-n(E\cap F)-n(F\cap G)-n(G\cap E)+n(E\cap F\cap G)$
$n(E\cup F\cup G)=31+23+20-18-11-13+8=40$
then students who doesn't know any language are
$n(E\cup F\cup G{)}^{\prime }=60-40=20$
students who knows English and German but not French are
$n(E\cap G)-n(E\cap F\cap G)=13-8=5$
Required number of students are $=20+5=25$