Directions: Study the following information carefully to answer the given questions: The teachers' colony has 2800 members, out of which 650 members read only English newspaper. 550 members read only Hindi newspaper and 450 members read only Marathi newspaper. The number of members reading all the 3 newspapers is 100. Members reading Hindi as well as English newspaper are 200. 400 members read Hindi as well as Marathi newspaper and 300 members read English as well as Marathi newspaper. How many members read at least two newspapers?

### Concept & Strategy "At least two" means we must sum the people who read *exactly two* newspapers and the people who read *all three* newspapers. This captures everyone who isn't a single-newspaper reader or a non-reader. $$\text{At least 2} = n(\text{Only } 2) + n(\text{All } 3)$$ ### Step-by-Step Solution * **Given Data:** * Total $H \cap E = 200$ * Total $H \cap M = 400$ * Total $E \cap M = 300$ * All three ($E \cap H \cap M$) = $100$ * **Calculation:** * First, isolate the distinct "Exactly Two" segments by subtracting the "All Three" overlap from each total pair. * Only $H \cap E = 200 - 100 = 100$ * Only $H \cap M = 400 - 100 = 300$ * Only $E \cap M = 300 - 100 = 200$ * Now, sum the "Exactly Two" groups and add the "All Three" group back in once. * At least two = (Only $H \cap E$) + (Only $H \cap M$) + (Only $E \cap M$) + (All 3) * At least two = $100 + 300 + 200 + 100 = 700$. * Since $700$ is not among the options (500, 600, 800, 1000), the answer is "None of these". ### Exam Strategy & Shortcut An elegant formula trick for "At least two" when given standard intersections is to sum all the dual intersections and then subtract twice the "all three" intersection. $$\text{At least 2} = n(A \cap B) + n(B \cap C) + n(A \cap C) - 2 \times n(A \cap B \cap C)$$ At least two = $(200 + 400 + 300) - 2(100) = 900 - 200 = 700$. This removes the need to find the individual "only two" pieces. ### Common Pitfall Students often just add up the intersections: $200 + 400 + 300 = 900$. This counts the center "all three" region *three times*. Because we only want to count those $100$ people once as part of the "at least two" group, we must subtract the extra two counts ($200$) from the total. ### Final Answer **Therefore, the correct answer is None of these.**