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What is the complete elaborated solution?

We are tasked with finding the probability that a triangle can be formed by selecting 3 points from a set of 16 points on a plane, where exactly 7 of those points are collinear.

### Key Concepts:
- **Collinear points** are points that lie on the same straight line, so no triangle can be formed using three collinear points.
- **Non-collinear points** can form a triangle.

### Step-by-Step Solution:

1. **Total number of points**: There are 16 points in total, with 7 of them being collinear.
   
2. **Choosing 3 points from 16**: The total number of ways to select 3 points from the 16 is given by the combination formula \( \binom{16}{3} \):
   \[
   \binom{16}{3} = \frac{16 \times 15 \times 14}{3 \times 2 \times 1} = 560
   \]
   So, there are 560 possible ways to choose 3 points from 16.

3. **Unfavorable outcomes (3 points from the collinear set)**: If we select all 3 points from the 7 collinear points, we will not be able to form a triangle. The number of ways to choose 3 points from these 7 collinear points is:
   \[
   \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
   \]
   So, there are 35 ways to choose 3 points that are all collinear.

4. **Favorable outcomes (forming a triangle)**: The favorable cases are those where at least one point is selected from the non-collinear points. Thus, the number of favorable ways to select 3 points that can form a triangle is:
   \[
   \text{Favorable outcomes} = \binom{16}{3} - \binom{7}{3} = 560 - 35 = 525
   \]

5. **Probability**: The probability of forming a triangle is the ratio of favorable outcomes to the total number of outcomes:
   \[
   P(\text{triangle}) = \frac{525}{560} = \frac{105}{112}
   \]
   Simplifying further:
   \[
   P(\text{triangle}) = \frac{15}{16}
   \]

### Final Answer:
The probability that a triangle can be formed by joining the points is \( \frac{15}{16} \), which corresponds to option **(a)**.

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Would you like further clarifications or additional details?

Here are 5 related questions you might find useful:
1. How can we generalize this method to any number of collinear points?
2. What if the number of collinear points was increased to 10? How would that affect the probability?
3. What is the probability of forming a quadrilateral from a set of points with some collinear?
4. Can we extend this approach to finding the probability of forming other shapes, like a rectangle or pentagon?
5. How does this method change if no points are collinear at all?

**Tip**: Always check for special conditions like collinearity when working with geometrical probability problems!

Suppose that 16 points are drawn on a plane such that exactly 7 of these points are collinear. Any set of three points which do not all belong to the 7 are noncollinear. If 3 random points are selected from the 16 points, what is the probability that a triangle can be formed by joining these points?

This problem involves calculating the probability that a triangle can be formed by randomly selecting 3 points from a set of 16 points, where 7 points are collinear. Using combinatorics and probability, learn how to solve this geometry-based problem.

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