In a joint family of 18 members, 10 take milk, 8 take tea and 6 take coffee; 5 take milk as well as tea but none takes milk and coffee. The number of members who take tea as well as coffee is

### Concept & Logic This is a standard 3-set Union problem simplified by a zero-overlap condition. Since "none takes milk and coffee" ($M \cap C = 0$), it is mathematically impossible for anyone to take all three ($M \cap T \cap C = 0$). We can use the standard union formula to solve for the missing intersection. $$n(M \cup T \cup C) = n(M) + n(T) + n(C) - n(M \cap T) - n(M \cap C) - n(T \cap C) + n(M \cap T \cap C)$$ ### Step-by-Step Solution * **Given:** * Total Family $= 18$ (Assume $n(M \cup T \cup C) = 18$) * Milk ($M$) = $10$, Tea ($T$) = $8$, Coffee ($C$) = $6$ * $M \cap T = 5$ * $M \cap C = 0$ * Since $M \cap C = 0$, logically $M \cap T \cap C = 0$ * Let $T \cap C = x$ * **Calculation:** * Substitute the values into the formula: * $18 = 10 + 8 + 6 - 5 - 0 - x + 0$ * $18 = 24 - 5 - x$ * $18 = 19 - x$ * $x = 19 - 18$ * $x = 1$ ### Exam Strategy & Shortcut When a 3-set problem tells you that two of the sets do not intersect, the universal union formula collapses into a much faster arithmetic string: Total = (Sum of Singles) - (Sum of provided intersections) - (Missing Intersection). $18 = (10+8+6) - 5 - x \Rightarrow 18 = 24 - 5 - x \Rightarrow x = 1$. This can be done entirely mentally in seconds. ### Common Pitfall Overthinking the "None" category. In questions phrasing like "In a family of 18 members, 10 take...", it is a standard assumption in aptitude exams that the universal set equals the union set (i.e., every member takes at least one beverage) unless explicitly stated otherwise. Assuming there is a hidden "None" value will make the equation unsolvable. ### Final Answer **Therefore, the correct answer is 1.**