Key Concepts and Formulas

• Combinations: The number of ways to choose $k$ objects from a set of $n$ distinct objects without regard to order is given by $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
• Complementary Counting: If it's difficult to directly count the number of objects with a certain property, count the total number of objects and subtract the number of objects without that property.
• Regular Polygon Properties: A regular $n$-sided polygon (or $n$-gon) has $n$ vertices and $n$ sides.

Step-by-Step Solution

Step 1: Calculate the Total Number of Possible Triangles

• What: We need to find the total number of triangles that can be formed by choosing any three vertices from the eight vertices of the octagon.
• Why: This forms the basis for complementary counting. We will subtract the "undesired" triangles from this total.
• Calculation: Using the combination formula, we have: $\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$
• Reasoning: Each combination of three vertices uniquely defines a triangle. Since order doesn't matter, we use combinations.

Step 2: Calculate the Number of Triangles with Exactly Two Sides Common with the Octagon

• What: We need to find the number of triangles formed by vertices of the octagon which share exactly two sides with the octagon.
• Why: These are the triangles formed by three consecutive vertices of the octagon.
• Calculation: The triangles are of the form $(V_1, V_2, V_3)$, $(V_2, V_3, V_4)$, ..., $(V_8, V_1, V_2)$, where $V_i$ represents the $i$-th vertex of the octagon. There are 8 such triangles. $\text{Triangles with 2 common sides} = 8$
• Reasoning: Each set of three consecutive vertices defines a unique triangle that shares two sides with the octagon. Since the octagon has 8 vertices, there are 8 such sets of consecutive vertices.

Step 3: Calculate the Number of Triangles with Exactly One Side Common with the Octagon

• What: We need to find the number of triangles that share exactly one side with the octagon.
• Why: We choose one side of the octagon and then choose a third vertex that is not adjacent to either vertex of the chosen side.
• Calculation:
  1. Choose one side of the octagon: There are 8 ways to do this.
  2. Suppose we choose side $V_iV_{i+1}$. The third vertex, $V_k$, cannot be $V_{i-1}$, $V_i$, $V_{i+1}$, or $V_{i+2}$. (Indices are taken modulo 8). This leaves $8 - 4 = 4$ choices for the third vertex. Therefore, the number of such triangles is $8 \times 4 = 32$. $\text{Triangles with 1 common side} = 8 \times 4 = 32$
• Reasoning: For each of the 8 sides of the octagon, we have 4 choices for the third vertex such that the resulting triangle has exactly one side in common with the octagon.

Step 4: Calculate the Number of Triangles with No Sides Common with the Octagon

• What: We use complementary counting to find the number of triangles with no sides common with the octagon.
• Why: We subtract the number of triangles with at least one side common with the octagon from the total number of triangles.
• Calculation: $\text{Desired Triangles} = \text{Total Triangles} - (\text{Triangles with 1 common side}) - (\text{Triangles with 2 common sides})$ $\text{Desired Triangles} = 56 - 32 - 8 = 16$
• Reasoning: We subtract the number of triangles with exactly one common side and the number of triangles with exactly two common sides from the total number of triangles.

Common Mistakes & Tips

• Overcounting/Undercounting: Carefully consider each case to avoid overcounting or undercounting. Drawing diagrams and labeling vertices can help.
• Adjacent vs. Non-Adjacent: Pay close attention to the definitions of "adjacent" and "non-adjacent" vertices or sides.
• Modulus Arithmetic: When dealing with polygons, remember to use modulus arithmetic for vertex indices (e.g., vertex $V_9$ in an octagon is actually vertex $V_1$).

Summary

We used complementary counting to find the number of triangles formed by the vertices of a regular octagon with no sides common with the octagon. We calculated the total number of triangles, the number of triangles with exactly one side common, and the number of triangles with exactly two sides common. Subtracting the latter two from the total gave us the answer of 16.

The final answer is $\boxed{16}$, which corresponds to option (B).