Question

In a survey of $55$ students, it was found that $25$ had taken Mathematics $22$ had taken Physics and $21$ had taken Chemistry, $12$ had taken Mathematics and Physics, $10$ had taken Mathematics and Chemistry and $8$ had taken Physics and Chemistry if $12$ students had taken none of the three subjects to find the number of students, who had taken all the three subjects. Also, find the number of those who have taken
(i) only Mathematics
(ii) only Physics
(iii) only Chemistry.

Answer 1

Let the set of students who took mathematics, physics and chemistry be represented by M, P and C respectively.

$n\left(U\right)=35$

$n\left(M\right)=25$

$n\left(P\right)=22$

$n\left(C\right)=21$

$n\left(MP\right)=12$

$n\left(MC\right)=10$

$n\left(PC\right)=8$

$n\left(none\right)=12$

$M\cup P\cup C=55-12=43$

$M\cup P\cup C=n\left(M\right)+n\left(P\right)+n\left(C\right)-n\left(MC\right)-n\left(PC\right)-n\left(MP\right)+n\left(MPC\right)$

$43=25+22+21-12-10-8+n\left(MPC\right)$

$n\left(MPC\right)=5$

Only mathematics and physics : $n\left(MP\right)-n\left(MPC\right)=7$

Only mathematics and chemistry : $n\left(MC\right)-n\left(MPC\right)=5$

Only physics and chemistry : $n\left(PC\right)-n\left(MPC\right)=3$

Only mathematics : $n\left(M\right)-7-5+n\left(MPC\right)=25-12+5=18$

Only physics :$n\left(P\right)-7-3+n\left(MPC\right)=22-10+5=17$

Only chemistry :$n\left(C\right)-5-3+n\left(MPC\right)=21-8+5=18$