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Question #41871

Difficulty: 8/10 | 🟢 Free

The Question

A permutation of the 13 letters of the word $\text{ASSASSINATION}$ is chosen uniformly at random. Let $X$ be the number of indices $i \in \{1, 2, \dots, 12\}$ such that the $i$ -th and $(i+1)$ -th letters are identical. Let $Y$ be the number of indices $j \in \{1, 2, \dots, 12\}$ such that both the $j$ -th and $(j+1)$ -th letters are vowels. Find the value of $E[X] + E[Y]$ .

Options

A.
$4$
B.
$\frac{52}{11}$
C.
$\frac{26}{7}$
D.
$\frac{48}{13}$

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

String Method (Adjacency)|Indicator Function Logic|Exhaustive Events Concept

Why This Question Is Tricky

Common Mistakes

The problem tests the application of Linearity of Expectation via Indicator Functions to bypass the 'Exhaustive Events' trap in complex permutation spaces.

Students often attempt to use the String Method (treating adjacent identical letters as a single block) combined with the Principle of Inclusion-Exclusion (PIE) to find the probability distribution of $X$ .

However, because letters like 'S' appear 4 times, the string logic requires partitioning into overlapping cluster sizes (e.g., SS, SSS, SSSS), making the exhaustive event space incredibly prone to over-counting errors.

By defining indicator variables for each adjacent pair, the expectation simplifies to a sum of marginal probabilities, completely circumventing the need for mutually exclusive event partitioning.

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