There are $100$ students in a class. In a test, $50$ of them failed in mathematics, $45$ failed in physics and $40$ failed in chemistry. $32$ failed in exactly two of these three subjects.Only one student passed in all the three subjects.The number of students failing in all three subjects is
My solution:
As only one student has passed in all three subjects so $99$ students have failed in at least one subject.
Denoting fail in mathematics as $M$, physics as $P$, chemistry as $C$. $MP$ denotes fail in math and phy. similarly $PC$ and $MC$. $MPC$ denote fail in all three subjects.
Number of students failed in $M$ OR $P$ OR $C$ = $M+P+C-MP-PC-MC+MPC$
Given that $32$ students failed exactly in two of these subjects. so $MP+PC+MC=32$.
$99=50+45+40-32+MPC$,
$MPC=-4$
Whats wrong here?
Help appreciated 🙂
Best Answer
It looks as if you are using the standard Inclusion/Exclusion formula. In that formula, $\text{MP}$ would represent the people who failed math and physics, and possibly chemistry. (The usual notation is something like $|M\cap P|$.) Similar remarks can be made about the other symbols.
The term $\text{MP}+\text{PC}+\text{MC}$ is then $32+3\text{MPC}$. This is because $\text{MPC}$ should be added three times to the count of people who failed exactly two subjects. So your equation should be $$99=50+45+40-(32+3\text{MPC})+\text{MPC}.$$