Question 1179262
Let's break down this problem step-by-step:

**1. Total Number of Segments:**

* With 10 points, the number of segments that can be formed is given by the combination formula:
    * ¹⁰C₂ = 10! / (2! * 8!) = (10 * 9) / 2 = 45

**2. Total Number of Ways to Choose 4 Segments:**

* The number of ways to choose 4 segments from the 45 available is:
    * ⁴⁵C₄ = 45! / (4! * 41!) = (45 * 44 * 43 * 42) / (4 * 3 * 2 * 1) = 148,995

**3. Number of Ways to Choose 4 Segments That DO NOT Form a Triangle:**

To find the probability of forming a triangle, it's easier to find the probability of *not* forming a triangle and subtract that from 1.

For 4 segments not to form a triangle, we need to consider the following cases:

* **Case 1: No intersections.** All four segments are completely separate.
* **Case 2: One intersection.** Two segments intersect at a point, but no triangle is formed.
* **Case 3: Two intersections.** Two sets of intersecting segments, but no triangle is formed.
* **Case 4: All 4 segments form a quadrilateral.**

We will use the complementary counting method.

Consider the cases where we *cannot* form a triangle:

* **Case 1: All 4 segments are disjoint.** This is difficult to calculate directly.
* **Case 2: Choose 4 segments such that no 3 form a triangle.**

Instead, let's look at the complementary case:

* **Case 1: 3 of the 4 segments form a triangle.**
    * Choose 3 points out of 10 to form a triangle: ¹⁰C₃ = 120
    * Choose 1 remaining segment: 42 segments can be chosen that don't form a triangle with the previous 3.
    * So, 120 * 42 = 5040 ways to have 3 segments form a triangle, but we overcount, so we must divide by the number of times we can select the same triangle.
* **Case 2: 4 segments form a complete quadrilateral.**
    * Choose 4 points out of 10: ¹⁰C₄ = 210
    * Each set of 4 points forms 3 possible quadrilaterals.
    * 210 * 3 = 630 ways to have 4 segments form a quadrilateral.

Let's use a different approach.

We need to subtract the cases where we cannot form a triangle.

* **Case 1: 4 disjoint segments.**
* **Case 2: 2 pairs of disjoint segments.**
* **Case 3: A "path" of 4 segments.**
* **Case 4: A "star" with 4 segments.**

Instead, let's find the number of ways to form a triangle.

* Choose 3 points out of 10: ¹⁰C₃ = 120
* Choose 1 more segment from the remaining 42 segments: 42
* Total ways to have a triangle: 120 * 42 = 5040

However, we are overcounting.
We can select the 3 segments of the triangle from the chosen 4 segments in 4 ways.

So we have to divide by 4.

5040/4 = 1260

So there are at least 1260 ways.

Probability = 1260/148995 = 84/9933 = 28/3311.

m=28, n=3311

m+n = 3339

**Final Answer:**

m + n = 28 + 3311 = 3339