Let $U$ be the universal set of students, $A$ be the set of students taking apple juice, and $O$ be the set of students taking orange juice. We are given $n(U) = 400$, $n(A) = 100$, $n(O) = 150$, and $n(A \cap O) = 75$. According to the principle of inclusion-exclusion, the number of students taking at least one juice is $n(A \cup O) = n(A) + n(O) - n(A \cap O) = 100 + 150 - 75 = 175$. The number of students taking neither juice is the complement of the union, $n(A' \cap O') = n((A \cup O)') = n(U) - n(A \cup O) = 400 - 175 = 225$. Option 2 is the number of students taking at least one juice. Option 3 and 4 represent common calculation errors such as neglecting the intersection or subtracting from a different base.