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Question #40616

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ and $L_2$ be two distinct parallel lines. Suppose $x$ distinct points are chosen on $L_1$ and $y$ distinct points are chosen on $L_2$ , where $x$ and $y$ are positive integers satisfying the constraint $2x + 3y \le 36$ . Let $T(x,y)$ denote the number of triangles that can be formed using these $x+y$ points as vertices. What is the maximum possible value of $T(x,y)$ ?

Options

A.
325
B.
336
C.
351
D.
356

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Geometry-based Combinations|Objective Function Gradient|Non-Negativity Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the 'Integer-Constraint vs.

Gradient-Continuity' trap in optimization.

Students often find the maximum of the continuous objective function using calculus (yielding x=10, y=5.33) and simply round to the nearest feasible integers (10,5) to get 325.

However, because the combinatorial objective function's gradient changes along the boundary, the true discrete maximum lies at a different lattice point (9,6) yielding 351.

Treating discrete variables as continuous without checking adjacent boundary lattice points is a fatal flaw in advanced LPP and combinatorics.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Geometry-based Combinations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free