*Given a regular 2n-gon, how many ways are there to pair vertices and draw line segments*

*between those vertices such that no two line segments intersect?*

This problem is a classic combinatorial geometry problem that involves pairing the vertices of a regular \(2n\)-gon such that no two line segments connecting the paired vertices intersect.

### Solution

Consider a regular \(2n\)-gon with vertices labeled as \(1, 2, 3, \ldots, 2n\). We want to find the number of ways to pair the vertices such that no two line segments (formed by connecting the paired vertices) intersect.

This problem is equivalent to finding the number of non-crossing perfect matchings of \(2n\) points arranged in a circle. This is a well-known combinatorial problem, and the number of such matchings is given by the \(n\)-th Catalan number, \(C_n\).

#### Catalan Number Formula

The \(n\)-th Catalan number \(C_n\) is given by:

\[
C_n = \frac{1}{n+1} \binom{2n}{n}
\]

where \(\binom{2n}{n}\) is a binomial coefficient.

#### Explanation

1. **Vertices and Pairings:** We need to find the number of ways to create \(n\) pairs of vertices such that no two line segments intersect. For a regular \(2n\)-gon, there are \(2n\) vertices.
   
2. **Non-Crossing Condition:** When pairing the vertices, we must ensure that no two chords intersect inside the polygon. This condition is satisfied by the Catalan number.

3. **Catalan Number Application:** The problem of finding non-intersecting chords is a well-known application of the Catalan number. Each valid pairing corresponds to a valid non-crossing matching of parentheses in the Catalan number's combinatorial interpretation.

#### Conclusion

The number of ways to pair vertices of a regular \(2n\)-gon and draw line segments between them such that no two line segments intersect is:

\[
\boxed{C_n = \frac{1}{n+1} \binom{2n}{n}}
\]

### Example

For \(n = 3\) (a regular hexagon, \(6\)-gon):

\[
C_3 = \frac{1}{3+1} \binom{6}{3} = \frac{1}{4} \times 20 = 5
\]

There are 5 ways to pair the vertices of a regular hexagon such that no two line segments intersect.

Would you like more details or have any questions?

### Related Questions

1. How does the Catalan number relate to different combinatorial structures, like binary trees or Dyck paths?
2. Can we derive the formula for Catalan numbers using a recursive approach?
3. How does this problem change if we consider intersecting line segments?
4. What is the connection between Catalan numbers and stack-based parsing algorithms?
5. How do Catalan numbers appear in different geometric configurations beyond polygons?

### Tip

Catalan numbers have a wide range of applications in combinatorics, from counting valid expressions in parentheses to describing the structure of certain types of trees and paths in grids. Understanding their different interpretations can provide deep insights into combinatorial mathematics!

Given a regular 2n-gon, how many ways are there to pair vertices and draw line segments between those vertices such that no two line segments intersect?

Discover how to calculate the number of ways to pair vertices of a regular 2n-gon and draw non-intersecting line segments between them. This problem involves Catalan numbers and combinatorics, providing insights into non-crossing matchings in geometry. Suitable for advanced high school and university students.

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