Venn diagram problem solving question In a class of $63$ students, $22$ study biology, $26$ study chemistry and $25$ study physics. $18$ study both physics and chemistry, $4$ study both biology and chemistry and $3$ study both physics and biology. One studies all three subjects. How many students study: a) biology only b) physics or chemistry c) physics but not chemistry

Let |$A$| denote the cardinality of $A$, and let $B,P,C$ denote the set of students studying biology,physics and chemistry respectively. Draw the Venn diagram, and you can see that:

a) for biology only, we have to delete the students in $B\cap C$ and $B\cap P$, but also have to add the students in $B\cup P\cup C$, since we are deleting it twice.

Therefore, number of students studying only biology $$= |B| - |B\cap C| - |B\cap P| + |B\cup P\cup C| = 22-4-3+1 = 16.$$

b) similarly, no. of students studying both physiics and chemistry is$$= |P\cup C|$$ $$=|P|+|C|-|P\cap C| = 25+26-18=33$$.

I hope you can do c) for yourself now. Please ask if you are stuck.