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

Answer

To solve the problem of how many triangles can be drawn using the vertices of a regular polygon with 20 sides, but not using the sides of the polygon, we will follow these steps:

### Step 1: Calculate the total number of triangles that can be formed using the vertices.
To find the total number of triangles that can be formed using the vertices of a polygon, we use the combination formula \( \binom{n}{r} \), where \( n \) is the number of vertices (or sides) and \( r \) is the number of vertices we want to choose to form a triangle. In this case, \( n = 20 \) and \( r = 3 \).

\[ \text{Total triangles} = \binom{20}{3} = \frac{20!}{3!(20-3)!} = \frac{20!}{3! \cdot 17!} \]

### Step 2: Simplify the combination.
We can simplify this expression:

\[ \binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = \frac{6840}{6} = 1140 \]

So, the total number of triangles formed by the vertices is 1140.

### Step 3: Calculate the triangles that use the sides of the polygon.
Next, we need to find the triangles that use at least one side of the polygon. We will consider two cases:

#### Case 1: Selecting 1 side.
If we select one side of the polygon, we can choose the third vertex from the remaining vertices. Since there are 20 sides, for each side, we can choose one of the 18 remaining vertices (excluding the two vertices of the selected side).

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

#### Case 2: Selecting 2 sides.
If we select two sides, they must be consecutive sides. For each pair of consecutive sides, there is only one possible vertex that can be selected as the third vertex, which is the vertex that is not adjacent to the two selected sides. There are 20 such pairs of consecutive sides.

\[ \text{Triangles using 2 sides} = 20 \]

### Step 4: Total triangles using sides.
Now, we can find the total number of triangles that use at least one side:

\[ \text{Total triangles using sides} = 360 + 20 = 380 \]

### Step 5: Calculate the triangles that do not use the sides.
Finally, we subtract the triangles that use the sides from the total number of triangles to find the triangles that do not use any sides:

\[ \text{Triangles not using sides} = \text{Total triangles} - \text{Total triangles using sides} = 1140 - 380 = 760 \]

### Final Answer:
The number of triangles that can be drawn using the vertices of the polygon, but not using the sides, is **760**.

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