A regular polygon has 20 sides. How many triangles can be drawn by using the vertices, but not using the sides?

Options

A. `.^(20)C_(1)..^(16)C_(2)`
B. `(.^(20)C_(1)..^(16)C_(2))/(3)`
C. 2400
D. 800

Answer

To find the number of triangles that can be formed using the vertices of a regular polygon with 20 sides, but not using any of the sides of the polygon, we can follow these steps:

### Step 1: Calculate the total number of triangles using vertices
We can choose any 3 vertices from the 20 vertices of the polygon to form a triangle. The number of ways to choose 3 vertices from 20 is given by the combination formula:

\[ \text{Total triangles} = \binom{20}{3} = \frac{20 \times 19 \times 18}{3!} = \frac{20 \times 19 \times 18}{6} \]

### Step 2: Simplify the combination
Calculating the above expression:

\[ \binom{20}{3} = \frac{20 \times 19 \times 18}{6} = \frac{6840}{6} = 1140 \]

So, there are 1140 triangles that can be formed using any 3 vertices.

### Step 3: Calculate the triangles that use the sides of the polygon
Next, we need to exclude the triangles that use the sides of the polygon. A triangle will use a side of the polygon if it consists of two adjacent vertices of the polygon and one other vertex.

For each side of the polygon, we can choose 1 additional vertex from the remaining vertices. Since there are 20 sides in the polygon, and for each side, we can choose 1 vertex from the remaining 18 vertices, the number of triangles that use the sides is:

\[ \text{Triangles using sides} = 20 \times \binom{18}{1} = 20 \times 18 = 360 \]

### Step 4: Calculate the triangles formed by 3 adjacent vertices
Additionally, we need to consider the triangles formed by 3 adjacent vertices (which are counted in the previous step). Each set of 3 adjacent vertices forms a triangle that uses the sides of the polygon. There are 20 such triangles (one for each set of 3 consecutive vertices).

### Step 5: Adjust for overcounting
Now, we need to adjust our count by subtracting the triangles that use the sides and adding back the triangles formed by 3 adjacent vertices:

\[ \text{Valid triangles} = \text{Total triangles} - \text{Triangles using sides} + \text{Triangles formed by 3 adjacent vertices} \]

Substituting the values we calculated:

\[ \text{Valid triangles} = 1140 - 360 + 20 = 800 \]

### Conclusion
Thus, the number of triangles that can be drawn using the vertices of a regular polygon with 20 sides, without using the sides of the polygon, is:

\[ \boxed{800} \]