Triangles are formed by joining vertices of an octagon. Any one of those triangle is selected at random. What is the probability that the selected triangle has no side common with the octagon?

Options

A. `3//7`
B. `2//7`
C. `5//7`
D. `1//7`

Answer

To solve the problem of finding the probability that a randomly selected triangle formed by the vertices of an octagon has no side common with the octagon, we will follow these steps:

### Step 1: Determine the total number of triangles that can be formed from the vertices of the octagon.

An octagon has 8 vertices. The number of ways to choose 3 vertices from these 8 to form a triangle is given by the combination formula:

\[ \text{Total triangles} = \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

### Step 2: Calculate the number of triangles that have at least one side in common with the octagon.

#### Case 1: Triangles with one side common
If we take one side of the octagon as common, we can choose any of the 8 sides. For each chosen side, we can select one of the remaining vertices (not adjacent to the chosen side) to form a triangle.

- For each side, there are 4 possible vertices to choose from (since the two vertices of the chosen side cannot be selected).
- Thus, the number of triangles with one side common is:

\[ \text{Triangles with one side common} = 8 \times 4 = 32 \]

#### Case 2: Triangles with two sides common
If we take two sides of the octagon as common, we can only form triangles using adjacent sides. There are 8 pairs of adjacent sides in the octagon.

- For each pair of adjacent sides, there is exactly 1 triangle that can be formed (the triangle formed by the two sides and the vertex opposite to them).
- Thus, the number of triangles with two sides common is:

\[ \text{Triangles with two sides common} = 8 \]

### Step 3: Calculate the total number of triangles with at least one side common.

Now, we can add the triangles with one side common and those with two sides common:

\[ \text{Total triangles with at least one side common} = 32 + 8 = 40 \]

### Step 4: Calculate the number of triangles with no sides common.

To find the number of triangles that have no sides in common with the octagon, we subtract the total number of triangles with at least one side common from the total number of triangles:

\[ \text{Triangles with no sides common} = \text{Total triangles} - \text{Total triangles with at least one side common} = 56 - 40 = 16 \]

### Step 5: Calculate the probability.

The probability that a randomly selected triangle has no side common with the octagon is given by the ratio of the number of triangles with no sides common to the total number of triangles:

\[ \text{Probability} = \frac{\text{Triangles with no sides common}}{\text{Total triangles}} = \frac{16}{56} = \frac{2}{7} \]

### Final Answer:
The probability that the selected triangle has no side common with the octagon is \(\frac{2}{7}\).

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