Fifteen persons are sitting around a circular table facing the centre. What is the probability that three particular persons sit together as a consecutive block?

Introduction / Context:
This question checks understanding of circular permutations and probability. We want the probability that three specific people, among fifteen seated around a circular table, sit together as neighbours, forming one block of three consecutive seats.

Given Data / Assumptions:

• Total people seated around the table: 15.
• All persons are distinct.
• Arrangements are around a circle with rotation considered the same arrangement.
• The three chosen persons must be seated consecutively, in any internal order, forming a single block.

Concept / Approach:
The number of distinct circular arrangements of n distinct persons is (n - 1)!. To count favourable arrangements we treat the three special persons as one combined block. This reduces the number of objects and changes the total circular permutations. Internal permutations of the block must also be counted. The probability equals favourable circular arrangements divided by total circular arrangements.

Step-by-Step Solution:
Total circular arrangements of 15 people = (15 - 1)! = 14!.Group the three particular persons into one block. Then we have this block plus 12 other persons, so 13 objects in total.Circular arrangements of these 13 objects = (13 - 1)! = 12!.Inside the block, the three persons can be arranged in 3! = 6 different orders.So number of favourable arrangements = 12! * 3!.Probability = (12! * 3!) / 14!.Simplify: 14! = 14 * 13 * 12!, so probability = 6 / (14 * 13) = 6 / 182 = 3 / 91.

Verification / Alternative check:
We can cross check by considering that the probability that the three specified persons occupy any chosen triple of consecutive seats is the same for all seat triples. There are 15 possible blocks of three consecutive seats around a 15 seat circle. Counting how many of these blocks the three persons can occupy and dividing by total circular permutations again leads to 3 / 91. This confirms the result.

Why Other Options Are Wrong:
The values 1/91, 2/73 and 3/73 arise from incorrect counting. For example, 1/91 suggests dividing by 13 * 7 instead of 14 * 13, and 2/73 or 3/73 come from using wrong total permutations or failing to account for the 3! internal arrangements of the block. None of these fractions matches the correct reduced form 3 / 91.

Common Pitfalls:
Students often forget that in circular arrangements, rotations are considered identical, so they mistakenly use 15! instead of 14! as the total. Another common error is to treat the three persons as fixed in one order, thus ignoring the 3! ways to rearrange them inside the block. These omissions significantly change the probability. Being systematic about total circular permutations and the block method helps avoid such mistakes.

Final Answer:
The required probability is 3/91.