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Question #41791

Difficulty: 8/10 | 🟢 Free

The Question

In a survey of students regarding their performance in three exams, it was found that 68 students passed Physics, 80 passed Chemistry, and 92 passed Mathematics. Let $x$ be the number of students who passed exactly two of these exams, and $y$ be the number of students who passed all three exams. A scholarship committee assigns a performance score $S$ to the group, given by $S = x + 12 \left\lfloor \frac{y}{5} \right\rfloor$ where $\lfloor \cdot \rfloor$ denotes the greatest integer function. Find the maximum possible value of $S$ .

Options

A.
168
B.
172
C.
174
D.
176

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Word Problem Modeling|Fundamental Principle of Counting (FPC)|Integral Part and Fractional Part|Cardinality Inclusion-Exclusion

Why This Question Is Tricky

Common Mistakes

This question tests the ability to model a word problem using the Principle of Inclusion-Exclusion to derive the hidden constraints of a feasible region ( $x+2y \le 148$ ).

The primary trap lies in the objective function containing a floor function, which forces non-linear discretization.

Students typically check only the vertices of the feasible region (e.g., $y=68 \implies S=168$ ), assuming the maximum of a linear-like model occurs at the boundaries.

However, the floor function's step-nature causes the true maximum to occur strictly inside an edge at a multiple of the denominator ( $y=65 \implies S=174$ ), rewarding analytical lattice-point checking over blind algorithm application.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Word Problem Modeling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free