In how many ways can ten points on a circle be joined to form five lines (each line joining a pair of points) in such a way that none of the lines cross each other?



The problem asks for the number of ways to join 10 points on a circle into 5 lines, such that none of the lines cross each other. This is a classic combinatorial problem related to *non-crossing pairings* or *non-crossing partitions*.

### Step-by-step solution:

1. **Understanding the structure**: 
   - We have 10 distinct points on a circle, and we are to form 5 non-crossing lines (each line joining 2 points). 
   - Since the points are arranged in a circle, we need to ensure that no lines intersect inside the circle. This leads us to consider the concept of *non-crossing partitions*.

2. **Non-crossing pairings and Catalan numbers**:
   - The number of ways to pair up \(2n\) points on a circle (without any pair of lines crossing) is given by the \(n\)-th **Catalan number** \(C_n\).
   - For this problem, \(n = 5\) because we are pairing up 10 points into 5 non-crossing lines.

3. **Catalan number formula**:
   The formula for the \(n\)-th Catalan number is:
   \[
   C_n = \frac{1}{n+1} \binom{2n}{n}
   \]
   Substituting \(n = 5\):
   \[
   C_5 = \frac{1}{5+1} \binom{10}{5} = \frac{1}{6} \times \binom{10}{5}
   \]

4. **Calculation**:
   First, calculate \(\binom{10}{5}\):
   \[
   \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252
   \]
   Now, compute \(C_5\):
   \[
   C_5 = \frac{1}{6} \times 252 = 42
   \]

Thus, the number of ways to join 10 points on a circle into 5 non-crossing lines is **42**.

Would you like more details or have any questions?

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Here are 5 related questions to consider:
1. How is the Catalan number related to other combinatorial structures?
2. What is the general formula for calculating a binomial coefficient?
3. How can we prove that the formula for the Catalan number leads to non-crossing pairings?
4. Can the Catalan numbers be applied to problems outside of geometry?
5. What are some real-world applications of non-crossing pairings?

**Tip**: Catalan numbers also appear in problems related to balanced parentheses and binary trees, so mastering them can help in many combinatorics problems!

In how many ways can ten points on a circle be joined to form five lines (each line joining a pair of points) in such a way that none of the lines cross each other?

Explore how to calculate the number of ways to join 10 points on a circle into 5 non-crossing lines using the Catalan number formula. This combinatorics problem applies the binomial coefficient and Catalan number theorem, making it ideal for Grades 10-12 students.

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