In a class of 50 students, 25 take Bengali, 16 take Hindi, 12 students take no language. How many take both Bengali and Hindi?

### Concept & Logic This is a fundamental 2-set Venn diagram problem. We use the Principle of Inclusion-Exclusion. First, find the "active" population (those who take at least one language), and then use the formula: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$. ### Step-by-Step Solution * **Given:** * Total students = 50. * Students taking no language = 12. * $n(Bengali) = 25$. * $n(Hindi) = 16$. * **Calculation:** First, determine how many students take *at least* one language. $$\text{Active Students} = \text{Total} - \text{None}$$ $$n(B \cup H) = 50 - 12 = 38$$ * Now apply the set formula to find the intersection (those taking both): $$n(B \cup H) = n(B) + n(H) - n(B \cap H)$$ $$38 = 25 + 16 - n(B \cap H)$$ $$38 = 41 - n(B \cap H)$$ * Solve for the intersection: $$n(B \cap H) = 41 - 38 = 3$$ ### Exam Strategy & Shortcut Add the individual subject totals and subtract the number of active students. Active students = $50 - 12 = 38$. Sum of subjects = $25 + 16 = 41$. The overlap (both) = Sum of subjects - Active students = $41 - 38 = 3$. You can do this entirely in your head without writing a single formula. ### Common Pitfall Forgetting to subtract the 12 students who take no language from the total 50. If you mistakenly use 50 as $n(A \cup B)$, you will get a negative number ($41 - 50$), which should immediately signal an error. ### Final Answer **Therefore, the correct answer is 3.**