Subject Choices Venn Diagram

Every student chooses at least two subjects, so region of exactly 1 subject is 0. Let regions be:

• $x$: Math and Physics only
• $y$: Math and Chem only
• $z$: Physics and Chem only
• All three = 18

Number of students choosing Math = $x + y + 18 = 23 \implies x + y = 5$. Number of students choosing Physics = $x + z + 18 = 25 \implies x + z = 7$. Number of students choosing Chem = $y + z + 18$. To minimize Chem students, we minimize $y + z$. $y + z = (5 - x) + (7 - x) = 12 - 2x$ To minimize $12 - 2x$, we maximize $x$. Since $x + y = 5$ and $x + z = 7$, max value of $x$ is 5 (since $y \ge 0$). So $x = 5 \implies y = 0, z = 2$. Minimum Chem students = $y + z + 18 = 0 + 2 + 18 = 20$.