Principle of Inclusion and Exclusion Problem Solving

Suppose that there are $100$ students in the class. Let $P$ and $M$ be the sets of students that passed the physics and math exams, respectively. Then $\lvert{P}\rvert=64,$ $\lvert{M}\rvert =68$ and $\lvert{P \cup M}\rvert \leq 100.$ Hence, $\begin{aligned} \lvert{P \cap M}\rvert &= \lvert{P}\rvert+\lvert{M}\rvert-\lvert{P \cup M}\rvert \\ &\geq 64+68-100 =32. \end{aligned}$ Hence, at least $32\text{\%}$ of the students passed both the physics and math exams.

$\text{physics} \colorbox{#20A900} {\hspace{64mm} 64\% } \colorbox{#CEBB00} {\hspace{36mm} 36\% } \\ \text{ maths } \colorbox{#CEBB00} {\hspace{32mm} 32\% } \colorbox{#20A900} {\hspace{68mm} 68\% } \\ \text{overlap} \colorbox{grey} {\hspace{32mm} 32\% }\colorbox{#3D99F6} {\hspace{22mm} 32\% } \colorbox{grey} {\hspace{37mm} 36\%}$

Therefore, minimum $32\%$ of the students passed both subjects.