In a class of 25 students, 12 have taken Mathematics, 8 have taken Mathematics but not Biology. The number of students who have taken both Mathematics and Biology is

### Concept & Logic This problem tests the relationship between a complete set and its distinct, non-overlapping parts. The total number of students taking a subject is the sum of those taking *only* that subject plus those taking *both* subjects. ### Step-by-Step Solution * **Given:** Total Mathematics students, $n(M) = 12$ Students taking Mathematics *only* (Math but not Biology), $n(M - B) = 8$ * **Calculation:** * The set of all Mathematics students is made of two distinct groups: those who take Math and Biology, and those who take Math *only*. $$n(M) = n(M \cap B) + n(M - B)$$ * We need to find the intersection, $n(M \cap B)$ (students taking both). Rearranging the formula: $$n(M \cap B) = n(M) - n(M - B)$$ * Substitute the values: $$n(M \cap B) = 12 - 8 = 4$$ ### Exam Strategy & Shortcut You don't need to look at the total class size (25) for this specific question. The phrase "12 have taken Mathematics" means the entire Math circle is 12. If the "Math ONLY" section is 8, the remainder of the Math circle must be the overlap. $12 - 8 = 4$. This can be solved in 5 seconds. ### Common Pitfall A common trap is trying to use the total class size (25) to find $n(B)$ (Biology students) using the union formula. This is a waste of time because the problem only asks for the intersection, which is fully contained within the Math data provided. ### Final Answer **Therefore, the correct answer is 4.**