Question

Twelve points are given in the plane, so that no three points lie on the same line.


Five distinct segments joining pairs of these points are chosen at random, with all \binom{12}{2} = 66 such segments equally likely to be chosen. What is the probability that a pentagon is formed by the chosen segments? (The pentagon's five vertices must all come from the original set of 12 points.)


Explain your answer.

Answer 1

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.

1. **Choose 5 vertices**: First, we select 5 points from the 12 points. The number of ways to choose 5 points is:

\[
\binom{12}{5}
\]

2. **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:

1. **Calculate \(\binom{12}{5}\)**:

\[
\binom{12}{5} = \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 792
\]

2. **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:

\[
\boxed{\frac{792}{8963958}} \text{ (approximately } 0.0000885 \text{)}
\]