The number of triangles whose vertices are a the vertices of an octagon but none of whose sides happen to come from the sides of the octagon is

Options

A. 16
B. 28
C. 56
D. 70

Answer

To solve the problem of finding the number of triangles whose vertices are the vertices of an octagon but none of whose sides happen to come from the sides of the octagon, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Total Number of Triangles from the Octagon:**
The total number of triangles that can be formed by selecting any three vertices from the eight vertices of an octagon is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of vertices and \( r \) is the number of vertices to choose.
\[ \text{Total triangles} = \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

2. **Calculate the Number of Triangles with At Least One Side from the Octagon:**
- **Triangles with Two Sides from the Octagon:**
A triangle can have two sides that are also sides of the octagon. This can happen if we select two adjacent vertices of the octagon and one other vertex. There are 8 pairs of adjacent vertices in an octagon, and for each pair, we can select one of the remaining 6 vertices.
\[ \text{Triangles with 2 sides from octagon} = 8 \times 6 = 48 \]

- **Triangles with One Side from the Octagon:**
A triangle can have one side that is a side of the octagon. If we select one side (which has 8 options), we can choose one of the remaining 6 vertices that are not adjacent to the selected side.
\[ \text{Triangles with 1 side from octagon} = 8 \times 4 = 32 \]

3. **Calculate the Total Number of Triangles with At Least One Side from the Octagon:**
We need to consider the overlap between triangles with one side and those with two sides. Since the triangles with two sides are already counted in the triangles with one side, we can simply add the two counts:
\[ \text{Total triangles with at least one side from octagon} = 48 + 32 - 8 = 72 \]
(Here, we subtract 8 because those triangles with two sides were counted twice.)

4. **Calculate the Number of Triangles with No Sides from the Octagon:**
To find the number of triangles that do not have any sides from the octagon, we subtract the number of triangles with at least one side from the total number of triangles:
\[ \text{Triangles with no sides from octagon} = 56 - 40 = 16 \]

### Final Answer:
The number of triangles whose vertices are the vertices of an octagon but none of whose sides happen to come from the sides of the octagon is **16**.

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