If three vertices of a regular hexagon are chosen at random, then the chance that they form an equilateral triangle is :

Options

A. `1/3`
B. `1/5`
C. `1/10`
D. `1/2`

Answer

To solve the problem of finding the probability that three randomly chosen vertices of a regular hexagon form an equilateral triangle, we can follow these steps:

### Step 1: Identify the total number of ways to choose 3 vertices from a hexagon.
A regular hexagon has 6 vertices. The number of ways to choose 3 vertices from 6 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Total ways to choose 3 vertices} = \binom{6}{3} \]

### Step 2: Calculate \( \binom{6}{3} \).
Using the combination formula:

\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 3: Identify the equilateral triangles that can be formed.
In a regular hexagon, the only sets of vertices that can form equilateral triangles are:
- Vertices 1, 3, and 5
- Vertices 2, 4, and 6

Thus, there are 2 equilateral triangles that can be formed.

### Step 4: Calculate the probability.
The probability \( P \) of choosing 3 vertices that form an equilateral triangle is given by the ratio of the number of favorable outcomes to the total outcomes.

\[ P = \frac{\text{Number of equilateral triangles}}{\text{Total ways to choose 3 vertices}} = \frac{2}{20} = \frac{1}{10} \]

### Final Answer:
Thus, the probability that three randomly chosen vertices of a regular hexagon form an equilateral triangle is \( \frac{1}{10} \).

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