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Question #41061
The Question
Let be the set of all possible circular arrangements of 4 married couples around a table such that no two men sit together. If an arrangement is chosen uniformly at random from , what is the probability that exactly two couples sit together (i.e., husband and wife occupy adjacent seats)?
Options
- A.
- B.
- C.
- D.
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
Equiprobability Fallacy|Sample Space Construction|Circular Permutations Constraint
Why This Question Is Tricky
Common Mistakes
Students often fall into the Equiprobability/Independence Fallacy by assuming that the events of each couple sitting together are independent.
Since a specific woman has 4 possible gaps and 2 of them are adjacent to her husband, the naive probability of a single couple sitting together is 2/4 = 1/2.
If one incorrectly assumes independence, the number of adjacent couples would follow a Binomial distribution B(4, 1/2), leading to the erroneous probability of C(4,2)(1/2)^2(1/2)^2 = 3/8.
The trap exploits the failure to recognize that the circular alternating constraint creates strong dependencies between the adjacencies of different couples, requiring rigorous use of the Principle of Inclusion-Exclusion to navigate the restricted sample space.
Quick Facts About This Question
Question Information
| Subject | Mathematics |
| Chapter | Probability - Class 11 |
| Concept | Equiprobability Fallacy |
| Difficulty | 8 |
| Cognitive Load | 7 |
| Target Solve Time | 160 sec |
| Access | Free |