Principle of inclusion and exclusion

Question

There are 250 students who failed in an examination. 128 failed in maths, 87 in physics, 134 in aggregate. 31 failed in maths and physics, 54 in aggregate and maths, 30 in aggregate and physics. Find out: 1) students failed in maths but not in physics 2) students failed in physics but not in aggregate or maths

Answer 1

HINT: Let $M,P$, and $A$ be the sets of students who failed in maths, physics, and aggregate, respectively. You’re told that $|M\cup P\cup A|=250$, $|M|=128$, $|P|=87$, $|A|=134$, $|M\cap P|=31$, $|M\cap A|=54$, and $|A\cap P|=30$, and you’re asked to find $|M\setminus P|$ and $|P\setminus(A\cup M)|$. It’s helpful to make a Venn diagram:

The red region is the set $M\setminus P$, and the blue region is the set $P\setminus(A\cup M)$. Finding the cardinality of the first is a matter of simply subtracting one of the known sizes from another. Finding the second is a little bit harder. I suggest first using an inclusion-exclusion calculation to find $|M\cup A|$; then recall that we know how many students failed altogether.