In an octagon triangles are formed by using vertices of octagon then number of triangles formed which has no side common with sides of octagon.

Options

A. 56
B. 48
C. 32
D. 16

Answer

To solve the problem of finding the number of triangles that can be formed using the vertices of an octagon without sharing any sides with the octagon, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to form triangles using the vertices of an octagon (8 vertices). However, the triangles must not share any sides with the octagon.

2. **Total Combinations of Triangles**:
First, we calculate the total number of triangles that can be formed using any three vertices from the octagon. This can be calculated using the combination formula:
\[ \text{Total triangles} = \binom{8}{3} \]
where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\).

3. **Calculating \(\binom{8}{3}\)**:
\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]
So, there are 56 triangles that can be formed using the vertices of the octagon.

4. **Subtracting Invalid Cases**:
Next, we need to subtract the triangles that have at least one side in common with the octagon.

- **Triangles with 1 side in common**:
If a triangle has one side in common with the octagon, we can choose any side of the octagon (there are 8 sides) and then choose one more vertex from the remaining vertices that do not form a side with the chosen side. For any side, we have 4 remaining vertices to choose from.
\[ \text{Triangles with 1 side in common} = 8 \times 4 = 32 \]

- **Triangles with 2 sides in common**:
If a triangle has two sides in common with the octagon, it must be formed by two adjacent vertices of the octagon and one vertex from the remaining vertices. There are 8 such pairs of adjacent vertices, and for each pair, we cannot use the two adjacent vertices, leaving us with 6 vertices to choose from.
\[ \text{Triangles with 2 sides in common} = 8 \times 1 = 8 \]

5. **Total Invalid Cases**:
Now we can find the total number of triangles that are invalid (i.e., those that share at least one side with the octagon):
\[ \text{Total invalid triangles} = 32 + 8 = 40 \]

6. **Calculating Valid Triangles**:
Finally, we subtract the invalid triangles from the total triangles:
\[ \text{Valid triangles} = \text{Total triangles} - \text{Total invalid triangles} = 56 - 40 = 16 \]

### Conclusion:
The number of triangles that can be formed using the vertices of the octagon without sharing any sides with the octagon is **16**.