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Question #41868

Difficulty: 8/10 | 🟢 Free

The Question

Let $N$ be the number of distinct permutations of the letters of the word "ADVANCED" such that all the vowels appear together. Let the sum $S$ be defined as $S = \sum_{k=1}^{N} \frac{k^2}{(k+1)!}$ Find the value of $\{ N! \cdot S \}$ , where $\{x\}$ denotes the fractional part function.

Options

A.
$\frac{1}{1081}$
B.
$\frac{1}{1080}$
C.
$\frac{1079}{1080}$
D.
$\frac{1080}{1081}$

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

String Method (Adjacency)|Sigma Notation Properties|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to combine combinatorial string methods with advanced telescoping series, culminating in a classic fractional part trap.

Students must correctly identify identical items in permutations to find N=1080.

The 'Lethal Collision' occurs when evaluating the fractional part of the product.

The series simplifies to an integer minus a fraction N/(N+1).

A common mistake is assuming the fractional part of a negative fraction -x is simply x, leading students to incorrectly choose 1080/1081.

The mathematical logic dictates that for a non-integer X = I - f (where 0 < f < 1), the fractional part is 1 - f, forcing the exact evaluation to 1/(N+1).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	String Method (Adjacency)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free