In a class of 50 students, 23 speak English, 15 speak Hindi and 18 speak Punjabi. 3 speak only English and Hindi, 6 speak only Hindi and Punjabi and 6 speak only English and Punjabi. If 9 speak only English, then how many speak all the three?

### Concept & Logic When you know the grand total of a single set (e.g., Total English speakers) and all its specific sub-components except one, you can find the missing component using simple subtraction. The "English" circle is composed of exactly four distinct parts. $$n(E) = n(\text{Only } E) + n(\text{Only } E \cap H) + n(\text{Only } E \cap P) + n(E \cap H \cap P)$$ ### Step-by-Step Solution * **Given Data for English Circle:** * Total English ($E$) = $23$ * Only English = $9$ * Only English and Hindi ($E \cap H$) = $3$ * Only English and Punjabi ($E \cap P$) = $6$ * Let the number of students who speak all three = $x$ * **Calculation:** * Sum the four distinct parts of the English circle and equate it to the total. * $23 = 9 + 3 + 6 + x$ * $23 = 18 + x$ * $x = 23 - 18$ * $x = 5$ ### Exam Strategy & Shortcut Ignore the distracting data! The prompt gives you the totals for Hindi ($15$), Punjabi ($18$), the total class ($50$), and "only Hindi and Punjabi" ($6$). None of this is needed. Immediately isolate the category where you have the most data—English—and perform a one-step subtraction: $23 - 9 - 3 - 6 = 5$. ### Common Pitfall Students often waste time trying to build out the entire 3-circle Venn diagram using the universal total ($50$). This leads to complex algebraic equations with multiple variables. Always look for a localized circle where all but one variable is provided. ### Final Answer **Therefore, the correct answer is 5.**