To determine the probability that a randomly chosen set of 5 line segments among the 12 points forms a pentagon, we first need to understand what it means for a set of segments to form a pentagon. A pentagon is formed when the segments connect 5 vertices in such a way that they connect each vertex exactly once without any segments crossing one another and without any vertices being connected to each other directly more than once.
Step 1: Total ways to choose 5 segments
We know there are 12 points and we want to choose 5 segments from the total segments formed by these points. The total number of segments formed from these 12 points is given by:
\[ \text{Total segments} = \binom{12}{2} = 66 \]
The number of ways to select 5 segments from these 66 segments is:
\[ \text{Ways to choose 5 segments} = \binom{66}{5} \]
Step 2: Count the successful configurations (forming a pentagon)
Next, we need to count how many ways we can select 5 segments that form a pentagon.
- Choose 5 vertices: First, we select 5 points from the 12 points. The number of ways to choose 5 points is:
\[ \binom{12}{5} \]
- Forming a pentagon: For the 5 chosen points, denoted as \( A, B, C, D, E \), we can connect them such that they form a simple pentagon. The connections must be \( (A, B), (B, C), (C, D), (D, E), (E, A) \). This specific configuration connects these 5 points in a cyclic manner, creating the required structure of a pentagon.
There is exactly one way to connect these 5 points into a pentagon once they are chosen.
Step 3: Total successful choices
Thus, the total number of successful outcomes (i.e., choosing segments that create a pentagon) is equal to the number of ways to choose the 5 vertices:
\[ \text{Successful choices} = \binom{12}{5} \]
Step 4: Calculate the probability
The probability \( P \) that a random choice of 5 segments forms a pentagon is given by the ratio of successful outcomes to total outcomes:
\[ P = \frac{\text{Successful choices}}{\text{Total choices}} = \frac{\binom{12}{5}}{\binom{66}{5}} \]
Calculating these terms:
- Calculate \(\binom{12}{5}\):
\[ \binom{12}{5} = \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 792 \]
- Calculate \(\binom{66}{5}\):
To compute \(\binom{66}{5}\):
\[ \binom{66}{5} = \frac{66 \cdot 65 \cdot 64 \cdot 63 \cdot 62}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \]
Calculating the numerator:
\[ 66 \cdot 65 = 4290, 4290 \cdot 64 = 274560, 274560 \cdot 63 = 17335080, 17335080 \cdot 62 = 1076774960 \]
Now, calculating the denominator \(5! = 120\):
\[ \binom{66}{5} = \frac{1076774960}{120} = 8963958 \]
Final Calculation of Probability
So, we can now plug these values back into our probability:
\[ P = \frac{792}{8963958} \approx 0.0000885 \]
Therefore, the final answer for the probability that a random selection of 5 segments from 12 points forms a pentagon is:
\[ \boxed{\frac{792}{8963958}} \text{ (approximately } 0.0000885 \text{)} \]
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