In an examination, $35\%$ candidates failed in one subject and $42\%$ failed in another subject while $15\%$ failed in both the subjects. If $2500$ candidates appeared at the examination, how many passed in either subject but not in both?

### Concept & Logic This question requires understanding regions in a Venn diagram. "Passed in either subject but not in both" means the student passed exactly one subject. If a student passes exactly one subject out of two, it logically means they failed exactly one subject. $$\text{Passed Exactly One} = \text{Failed Exactly One}$$ ### Step-by-Step Solution * **Given:** * Total candidates = $2500$ * Failed in Subject 1, $n(A) = 35\%$ * Failed in Subject 2, $n(B) = 42\%$ * Failed in both, $n(A \cap B) = 15\%$ * **Calculation / Deduction:** * Students failing ONLY in Subject 1 = $35\% - 15\% = 20\%$. * Students failing ONLY in Subject 2 = $42\% - 15\% = 27\%$. * Total students failing exactly one subject = $20\% + 27\% = 47\%$. * As established, failing exactly one subject is logically identical to passing exactly one subject (passing either but not both). * Therefore, $47\%$ of the students passed exactly one subject. * Number of candidates = $47\%$ of $2500$ * $0.47 \times 2500 = 47 \times 25 = 1175$. ### Exam Strategy & Shortcut To find the symmetric difference (regions unique to each set without the intersection), you can directly use the formula: $n(A) + n(B) - 2 \cdot n(A \cap B)$. Here, it is $35\% + 42\% - 2(15\%) = 77\% - 30\% = 47\%$. Then directly compute $47 \times 25$ using the split multiplication method: $50 \times 25 - 3 \times 25 = 1250 - 75 = 1175$. ### Common Pitfall Students often get confused by the phrasing "passed in either subject but not in both". They attempt to calculate the passing percentages for each subject first and draw a new diagram, which takes much longer. Recognize that "passing exactly one" targets the exact same population as "failing exactly one." ### Final Answer **Therefore, the correct answer is 1175.**