Venn Diagram of Beverage Preferences

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 $x_{\text{max}} = 52$.

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 $x_{\text{min}} = 5$.

Difference between maximum and minimum $= 52 - 5 = 47$.