In a group of 20 persons there are exactly 2 brothers. In how many different circular arrangements of all 20 persons around a round table will there be exactly one person sitting between the two brothers?

Introduction / Context:
This problem involves circular permutations with a restriction on the relative positions of two specific people. Circular arrangement questions are more subtle than linear arrangements because rotations of a configuration are considered the same. Here, we want exactly one person between the two brothers, so we must respect both circular symmetry and the spacing constraint. Such questions appear in higher level aptitude and reasoning tests.

Given Data / Assumptions:

• Total persons: 20, including 2 brothers and 18 other distinct people.
• They are seated around a circular table.
• Arrangements that differ only by rotation are considered the same circular arrangement.
• There must be exactly one person between the two brothers.
• The direction (clockwise or anticlockwise) around the table is fixed for counting, but we account for circular symmetry properly.

Concept / Approach:
For circular permutations of n distinct people, the number of unrestricted arrangements is (n - 1)!. To handle the constraint, we use a fixing technique. We fix the position of one brother to break rotational symmetry, then decide positions for the second brother consistent with exactly one person between them, and finally permute the remaining 18 people freely in the remaining seats.

Step-by-Step Solution:
Step 1: Fix Brother 1 at a reference position on the circle to remove rotational duplicates. Step 2: With 20 seats in total, once Brother 1 is fixed, there are 19 remaining seats. Step 3: Brother 2 must sit with exactly one person between him and Brother 1. Step 4: On a circle, there are exactly two such seats for Brother 2: one two steps clockwise from Brother 1 and one two steps anticlockwise. Step 5: Therefore, there are 2 choices for Brother 2 once Brother 1 is fixed. Step 6: After seating both brothers, 18 remaining distinct people can be arranged in the remaining 18 seats in 18! ways. Step 7: Total valid arrangements = 2 * 18!.

Verification / Alternative check:
If there were no restriction, the number of circular arrangements of 20 distinct people would be 19!. Our answer 2 * 18! is equal to (2 / 20) * 19!, so about one tenth of all unrestricted circular arrangements satisfy the condition that the brothers have exactly one person between them. This ratio is plausible since there are many possible relative positions for two people around a large circle, and only two of them produce exactly one person between them, once rotational symmetry is accounted for.

Why Other Options Are Wrong:
18! x 19!: This multiplies two factorial counts incorrectly and does not reflect circular symmetry. 2 x 19!: This would be correct if we ignored circular fixing and overcounted rotations. 18! x 18!: A product with no correct combinatorial interpretation for this setup. Only 2 x 18! correctly follows from fixing one brother, choosing two possible seats for the other, and permuting the rest.

Common Pitfalls:
A major pitfall is forgetting that circular arrangements have rotational symmetry, which should be handled by fixing one person in place. Another is miscounting the number of valid positions for the second brother; some students think there are more than two such seats without visualizing the circle properly. It is also easy to accidentally use 19! directly rather than 2 * 18!, or to treat the arrangement as linear, which changes the count dramatically. Drawing a rough circle with labeled positions can help clarify the reasoning.

Final Answer:
The total number of circular arrangements in which exactly one person sits between the two brothers is 2 x 18!.