To find the number of ways to select 3 persons from n persons sitting around a circular table such that no two selected persons are neighbors, we follow these steps:
Step 1: Convert the Circular Selection into a Linear Form
Since the people are seated in a circle, we can first fix one person and then count the number of ways to choose the remaining two, ensuring no two selected persons are adjacent.
Step 2: Transformation into a Linear Selection Problem
Instead of selecting directly from n persons, we introduce a gap of at least one person between any two selected persons. To simplify this, we imagine replacing the selected persons with "chosen spots," and the non-selected persons with "gaps" between them.
If we select 3 persons, then there are (n - 3) persons left, and these create the necessary gaps between the chosen ones.
Step 3: Reducing the Problem to a Combination with Constraints
We now redefine the problem as selecting 3 persons from (n - 3) non-adjacent positions, which is equivalent to choosing 3 elements from a set of (n - 3) positions.
- To do this, we first place n persons in a row and ensure that between every two selected persons, at least one person is left out.
- To count this correctly, we transform the selection into a combinatorial problem.
Using the formula for non-adjacent selections, the number of ways to select 3 persons is:
(1/6) * n * (n - 4) * (n - 5)
Conclusion:
Thus, the number of ways to choose 3 persons such that no two are neighbors is:
(1/6) * n * (n - 4) * (n - 5)
