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Question #41886

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all permutations of the numbers $\{1, 2, 3, 4, 5, 6\}$ . For any permutation $\pi \in S$ , let $X(\pi)$ be the number of indices $i \in \{1, 2, 3, 4, 5\}$ such that $\pi(i+1) - \pi(i) = 1$ . Let $p_k$ be the probability that a randomly chosen permutation $\pi \in S$ satisfies $X(\pi) = k$ . Find the value of $\sum_{k=0}^{5} \left\lfloor \frac{720 \cdot p_k}{k+1} \right\rfloor$ where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ .

Options

A.
484
B.
485
C.
486
D.
487

Check Your Answer

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

Exhaustive Events Concept|String Method (Adjacency)|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question creates a 'Lethal Collision' by intertwining the String Method for exhaustive events with the non-linear floor function.

Students often calculate the total expected value or sum the exact fractional terms before applying the floor function, incorrectly assuming $\sum \lfloor x_i \rfloor = \lfloor \sum x_i \rfloor$ .

By forcing a partition based on the floor of each discrete probability state, the fractional parts (0.5, 0.66, 0.5, 0.166) are discarded individually.

This invalidates intuitive symmetry and aggregate summation tricks, demanding rigorous inclusion-exclusion counting for each boundary condition.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Exhaustive Events Concept
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free