Question

In a group of 150 students, 52 like tea, 48 like juice, and 62 like coffee. If each student in the group likes at least one among tea, juice, and coffee, then the maximum number of students that like more than one drink is _______________.

Hint:

When using inclusion-exclusion, check for maximum overlap conditions to maximize the number of students who like more than one drink.

Answer 1

 

1. Define the Sets and Use Inclusion-Exclusion:
   Let \( T \), \( J \), and \( C \) represent the sets of students who like tea, juice, and coffee, respectively, with:
   \( |T| = 52 \), \( |J| = 48 \), \( |C| = 62 \), and the total number of students \( |T \cup J \cup C| = 150 \).
   Each student likes at least one drink, so the union of the sets covers all students. Using the Principle of Inclusion-Exclusion:

\[ |T \cup J \cup C| = |T| + |J| + |C| - |T \cap J| - |T \cap C| - |J \cap C| + |T \cap J \cap C| \]

1. Substituting the known values:

\[ 150 = 52 + 48 + 62 - |T \cap J| - |T \cap C| - |J \cap C| + |T \cap J \cap C| \]

\[ 150 = 162 - (|T \cap J| + |T \cap C| + |J \cap C|) + |T \cap J \cap C| \]

1. Let \( S_2 = |T \cap J| + |T \cap C| + |J \cap C| \) (sum of pairwise intersections) and \( S_3 = |T \cap J \cap C| \). Then:

\[ 150 = 162 - S_2 + S_3 \]

\[ S_2 - S_3 = 162 - 150 = 12 \]

1. Maximize Students Liking More Than One Drink:
   Students liking more than one drink are those in at least two sets, i.e., \( (T \cap J) \cup (T \cap C) \cup (J \cap C) \). The number of such students is:

\[ |T \cap J| + |T \cap C| + |J \cap C| - 3|T \cap J \cap C| + |T \cap J \cap C| = S_2 - 2S_3 \]

1. We need to maximize \( S_2 - 2S_3 \) subject to \( S_2 - S_3 = 12 \), \( S_3 \geq 0 \), and the individual intersections being non-negative integers satisfying set constraints (e.g., \( |T \cap J| \leq \min(|T|, |J|) = 48 \)).
   • From \( S_2 - S_3 = 12 \), we have \( S_2 = S_3 + 12 \).
   • Substitute into the expression to maximize: \( S_2 - 2S_3 = (S_3 + 12) - 2S_3 = 12 - S_3 \).
   • To maximize \( 12 - S_3 \), minimize \( S_3 \). Since \( S_3 \geq 0 \), the minimum is \( S_3 = 0 \).
2. Verify Feasibility:
   Assume \( S_3 = |T \cap J \cap C| = 0 \). Then \( S_2 = |T \cap J| + |T \cap C| + |J \cap C| = 12 \). Assign, for example:
   • \( |T \cap J| = 12 \), \( |T \cap C| = 0 \), \( |J \cap C| = 0 \).

Answer: The maximum number of students who like more than one drink is \( 12 \).