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Question #40590

Difficulty: 8/10 | 🟢 Free

The Question

A researcher is distributing $6$ distinct energy modules and $14$ identical cooling units into $4$ distinct server racks. Each rack must receive at least $1$ energy module and at least $2$ cooling units to function properly. Due to spatial limitations, no server rack can contain more than $10$ items in total (energy modules and cooling units combined). Find the total number of valid distributions.

Options

A.
131040
B.
130560
C.
130080
D.
128640

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Division into Groups|Distributing Identical Items (Stars and Bars)|Redundant Constraints

Why This Question Is Tricky

Common Mistakes

This question employs a 'Redundant Constraint Masking' trap.

The condition $t_i \le 10$ appears to require complex inclusion-exclusion because it couples the distinct and identical distributions ( $d_i + c_i \le 10$ ).

Students who attempt to solve this forward will face a combinatorial explosion of cases for $d_i$ and $c_i$ .

The logical leap is to analyze the macroscopic boundary: since the minimum capacity of any box is 3, the absolute maximum any box can hold is 11.

Thus, the 'complex' constraint only excludes a single boundary state ( $t_i = 11$ ), turning a massive calculation into a brief logical deduction.

Quick Facts About This Question

Question Information

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