Question

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

Accepted Answer

C. Let $C, T, L$ be the sets of students liking coffee, tea, and lemonade. $|C| = 73, |T| = 80, |L| = 52, N = 100$. Let $x$ be the number of students who like all three drinks. **Maximum possible value of $x$:** $x \le \min(|C|, |T|, |L|) = \min(73, 80, 52) = 52$. So $x_{	ext{max}} = 52$. **Minimum possible value of $x$:** Using the principle of inclusion-exclusion: $$|C \cup T \cup L| = |C| + |T| + |L| - (|C \cap T| + |T \cap L| + |L \cap C|) + x \le 100$$ Also, $|C \cap T| + |T \cap L| + |L \cap C| \le |C| + |T| + |L| - x = 205 - x$. More directly, for any set of size $N$: $$x \ge |C| + |T| + |L| - 2N = 73 + 80 + 52 - 200 = 5$$ So $x_{	ext{min}} = 5$. Difference between maximum and minimum $= 52 - 5 = 47$.

Venn Diagram of Beverage Preferences

Explanation

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