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. Find the number of members reading no newspaper.

### Concept & Strategy To find the number of people who read *no* newspaper, we must subtract the total number of people who read *at least one* newspaper (the Union of all three sets) from the total population of the colony. $$\text{None} = \text{Total Population} - n(E \cup H \cup M)$$ ### Step-by-Step Solution * **Given Data:** * Total Population = $2800$ * Only English ($E$) = $650$, Only Hindi ($H$) = $550$, Only Marathi ($M$) = $450$ * All three ($E \cap H \cap M$) = $100$ * Total $H \cap E = 200$, Total $H \cap M = 400$, Total $E \cap M = 300$ * **Step 1: Find the "Only Two" distinct regions.** * Subtract the "All three" overlap from the total dual overlaps. * Only $H \cap E = 200 - 100 = 100$ * Only $H \cap M = 400 - 100 = 300$ * Only $E \cap M = 300 - 100 = 200$ * **Step 2: Calculate Total Readers (The Union).** * Sum all the completely distinct, non-overlapping regions inside the Venn diagram. * Total Readers = (Sum of "Only One") + (Sum of "Only Two") + ("All Three") * Total Readers = $(650 + 550 + 450) + (100 + 300 + 200) + 100$ * Total Readers = $1650 + 600 + 100 = 2350$ * **Step 3: Calculate Non-Readers.** * Non-Readers = Total Population - Total Readers * Non-Readers = $2800 - 2350 = 450$ * Because $450$ is not listed in options a, b, c, or d, the correct choice is "None of these". ### Exam Strategy & Shortcut Instead of isolating the "Only Two" regions, you can use the expanded union formula where you add the "Only One" regions to the total intersections and then adjust for the center: Total Union = (Only E + Only H + Only M) + $(H \cap E) + (H \cap M) + (E \cap M) - 2 \times (\text{All 3})$. Union = $1650 + (200 + 400 + 300) - 2(100) = 1650 + 900 - 200 = 2350$. None = $2800 - 2350 = 450$. ### Common Pitfall A very common mistake is assuming that "Members reading Hindi as well as English are 200" means *only* Hindi and English. If you don't subtract the $100$ who read all three, you will double-count the center, inflate the union to $2650$, and incorrectly conclude that only $150$ people read no newspaper (Option a). ### Final Answer **Therefore, the correct answer is None of these.**