In a group of 52 persons, 16 drink tea but not coffee and 33 drink tea. How many drink coffee but not tea?

### Concept & Logic This problem requires breaking down two sets into three distinct zones: Set A only, Set B only, and Both. By definition, the total group (Union) is the sum of these three non-overlapping zones: $(A \text{ only}) + (B \text{ only}) + (\text{Both})$. ### Step-by-Step Solution * **Given:** Total group size (assuming everyone drinks at least one), $n(T \cup C) = 52$ Tea only, $n(T - C) = 16$ Total Tea, $n(T) = 33$ * **Calculation:** * First, let's find how many people drink *both*. We know the total Tea drinkers consist of "Tea only" plus "Both". $$n(T) = n(T - C) + n(T \cap C)$$ $$33 = 16 + n(T \cap C)$$ $$n(T \cap C) = 33 - 16 = 17$$ * Now, we want to find "Coffee only", which is $n(C - T)$. We know the total group is 52. $$\text{Total} = (Tea \text{ only}) + (\text{Both}) + (Coffee \text{ only})$$ $$52 = 16 + 17 + n(C - T)$$ $$52 = 33 + n(C - T)$$ * Solve for Coffee only: $$n(C - T) = 52 - 33 = 19$$ ### Exam Strategy & Shortcut Notice that the total Tea circle $n(T)$ perfectly encompasses both "Tea only" and "Both". Therefore, if you subtract the entire Tea circle from the total population, you are left exclusively with "Coffee only". $$\text{Coffee Only} = \text{Total Population} - \text{Total Tea Drinkers}$$ $$\text{Coffee Only} = 52 - 33 = 19$$ You do not even need the "16 drink tea but not coffee" data point—it is entirely redundant filler! ### Common Pitfall Getting distracted by the redundant "16 drink tea but not coffee" and spending time calculating the intersection (17), only to forget what the final goal is and mistakenly selecting 17 (Option c) as the final answer. Always keep your eye on the specific metric requested. ### Final Answer **Therefore, the correct answer is 19.**