Problem: Number of Triangles with No Side Common with Heptagon

Given: A heptagon (a polygon with seven sides)

To find: The number of triangles that can be formed using the vertices of the heptagon, such that none of the sides of the triangle are common with any side of the heptagon.

Explanation:

Step 1: Understanding the Problem
A heptagon has seven vertices. We need to form triangles using these vertices. However, the triangles formed should not have any side common with the heptagon.

Step 2: Identifying the Conditions
To form a triangle with no side common with the heptagon, the triangle's three vertices should not be adjacent to each other in the heptagon.

Step 3: Counting the Triangles
Let's analyze the possible triangles that can be formed:

- If we choose any vertex of the heptagon, there are six possible vertices remaining to form the triangle.
- However, we need to exclude the two vertices adjacent to the chosen vertex in the heptagon. Therefore, the number of possible vertices is reduced to four.
- Now, if we choose one of these four vertices, there are three possible vertices remaining to form the triangle.
- Again, we need to exclude the two vertices adjacent to the chosen vertex. Therefore, the number of possible vertices is reduced to one.
- Finally, we can choose any of the three remaining vertices to form the triangle.

Therefore, the number of possible triangles that can be formed is 6 × 4 × 3 = 72.

Step 4: Eliminating Overcounting
In the above calculation, we have counted each triangle six times because each triangle has six possible starting vertices.

To eliminate this overcounting, we divide the total count by 6.

Therefore, the number of triangles with no side common with the heptagon is 72 ÷ 6 = 12.

Step 5: Checking the Options
The given options are:
a) 7
b) 14
c) 21
d) 28

None of the options match our result of 12.

Therefore, none of the options is correct.

However, we have made a mistake in our calculation, as the correct answer should be among the given options.

Step 6: Correct Calculation
Let's reanalyze the possible triangles that can be formed:

- If we choose any vertex of the heptagon, there are six possible vertices remaining to form the triangle.
- However, we need to exclude the two vertices adjacent to the chosen vertex in the heptagon. Therefore, the number of possible vertices is reduced to four.
- Now, if we choose one of these four vertices, there are three possible vertices remaining to form the triangle.
- Again, we need to exclude the two vertices adjacent to the chosen vertex. Therefore, the number of possible vertices is reduced to one.
- Finally, we can choose any of the three remaining vertices to form the triangle.

Therefore, the number of possible triangles that can be formed is 6 × 4 × 3 = 72.

However, we have overcounted the triangles again, as each triangle has three possible starting vertices.

Therefore, we need to divide the total count by

A heptagon has 7 vertices.