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Question #40564

Difficulty: 8/10 | 🟢 Free

The Question

Consider all permutations of the letters of the word $TARTAR$ . For any permutation, let $x_k$ denote the alphabetical rank of the letter at the $k$ -th position ( $1 \le k \le 6$ ), where $A=1, R=2, T=3$ . A permutation is defined as \textit{valid} if it satisfies the condition: $\sum_{k=1}^3 \sin^2\left(\frac{\pi}{2}(x_k - x_{k+3})\right) \neq 0$ Find the rank of the word $RATTAR$ among all valid permutations when they are arranged in dictionary order.

Options

A.
32
B.
35
C.
38
D.
41

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

Permutations of Identical Objects|Rank of a Word|Symmetry and Periodicity

Why This Question Is Tricky

Common Mistakes

This question fuses lexicographical rank calculation with a trigonometric periodicity constraint, creating a sophisticated Inclusion-Exclusion trap.

Students typically calculate the rank of a word by rigidly filling slots.

However, the condition $\sin^2(\frac{\pi}{2}\Delta x) = 0$ forces them to identify and subtract 'forbidden' permutations (where letter ranks differ by an even integer).

The primary trap lies in assuming the forbidden condition only implies strict periodicity ( $L_k = L_{k+3}$ ), which yields only 6 words.

By using alphabetical ranks, the difference between 'A' (1) and 'T' (3) is 2 (an even number), introducing 12 additional pseudo-periodic permutations.

Failing to account for these mixed $(A,T)$ pairs leads to an under-subtraction and an incorrect rank of 38.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Permutations of Identical Objects
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free