Out of 100 students in a class, 60 take tea, 40 take coffee and 25 take both. The number of students not taking either tea or coffee is

### Concept & Logic This is a classic Set Theory problem solvable using the Principle of Inclusion-Exclusion. To find the number of students taking *neither* beverage, we first need to determine how many students take *at least one* beverage (the union of both sets), and subtract that from the total class size. ### Step-by-Step Solution * **Given:** Total students = 100 Students taking tea, $n(T) = 60$ Students taking coffee, $n(C) = 40$ Students taking both, $n(T \cap C) = 25$ * **Calculation:** * First, find the total number of students taking at least one beverage using the union formula: $$n(T \cup C) = n(T) + n(C) - n(T \cap C)$$ * Substitute the given values: $$n(T \cup C) = 60 + 40 - 25$$ $$n(T \cup C) = 100 - 25 = 75$$ * This means 75 students take either tea, coffee, or both. * Now, find the students taking *neither*: $$\text{Neither} = \text{Total} - n(T \cup C)$$ $$\text{Neither} = 100 - 75 = 25$$ ### Exam Strategy & Shortcut **Draw a quick mental Venn Diagram:** The "both" section is 25. The "only tea" section is $60 - 25 = 35$. The "only coffee" section is $40 - 25 = 15$. Total active drinkers = $35 (\text{only tea}) + 15 (\text{only coffee}) + 25 (\text{both}) = 75$. Total students are 100, so $100 - 75 = 25$ students drink nothing. ### Common Pitfall The most common mistake is assuming that $n(T) + n(C)$ equals the total number of drinkers ($60 + 40 = 100$). This double-counts the students who drink both. Always subtract the intersection when combining totals. ### Final Answer **Therefore, the correct answer is 25.**