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Question #40319

Difficulty: 8/10 | 🟢 Free

The Question

Let $N$ be the number of ways to distribute 6 distinct objects into 3 identical boxes such that no box remains empty. Let $M$ be the number of ways to distribute the same 6 distinct objects into 3 distinct boxes such that no box remains empty. Evaluate the sum $\sum_{k=1}^{N} \left[ \frac{k \cdot M}{N^2} + \frac{1}{2} \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$ .

Options

A.
231
B.
273
C.
285
D.
369

Check Your Answer

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

Cardinality Inclusion-Exclusion|Division into Groups|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the Inclusion-Exclusion principle for surjective functions (distributing distinct objects into distinct boxes) and adjusting for identical boxes by dividing by $r!$ .

The primary trap lies in the Greatest Integer Function (GIF) summation.

If a student forgets the inclusion-exclusion principle and assumes $M = 3^6 = 729$ , the fractional multiplier changes entirely, leading to a completely different integer sum (369).

Furthermore, omitting the $+1/2$ shift results in 231.

The GIF acts as a strict hash function, amplifying any combinatorial miscalculation into a catastrophic jump-discontinuity error.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Cardinality Inclusion-Exclusion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free