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Question #40609

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be a multiset of letters $A_0, A_1, A_2, A_3$ where the frequency of each letter $A_k$ is given by $\binom{3}{k}$ for $k=0, 1, 2, 3$ . All possible words formed by permuting the letters of $S$ are arranged in dictionary order (assuming the alphabetical order is $A_0 < A_1 < A_2 < A_3$ ). If a word is chosen at random from this arrangement, what is the probability that it appears strictly before the word $W = A_2 A_2 A_1 A_1 A_1 A_2 A_0 A_3$ ?

Options

A.
$\frac{5}{8}$
B.
$\frac{11}{16}$
C.
$\frac{3}{4}$
D.
$\frac{21}{32}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Rank of a Word|Properties of Binomial Coefficients|Combinatorial Probability

Why This Question Is Tricky

Common Mistakes

This problem tests the student's ability to navigate a restricted state space in combinatorial probability.

The trap lies in the dual-layer counting: first, recognizing the multinomial denominator generated by binomial coefficient frequencies, and second, meticulously calculating the lexicographical rank without missing intermediate permutations.

Students often miscalculate the permutations of multisets when prefixes consume repeated letters, leading to an incorrect numerator, or they fail to account for the 'strictly before' condition.

Quick Facts About This Question

Question Information

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