Consider a regular polygon with 10 sides, What is the number of triangles that can be formed by joining the vertices which have no common side with any of the sides of the polygon?

Options

A. 25
B. 50
C. 75
D. 100

Answer

To solve the problem of finding the number of triangles that can be formed by joining the vertices of a regular polygon with 10 sides, ensuring that no triangle shares a side with the polygon, we can follow these steps:

### Step 1: Calculate the total number of triangles that can be formed from the vertices of the polygon.

We have a polygon with 10 vertices. The number of ways to choose 3 vertices from 10 is given by the combination formula:

\[ \text{Total triangles} = \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3! \cdot 7!} \]

### Step 2: Simplify the combination.

Calculating this gives:

\[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120 \]

So, there are 120 triangles that can be formed using any three vertices of the polygon.

### Step 3: Exclude triangles that share a side with the polygon.

#### a. Triangles using one side of the polygon:

If we take one side of the polygon, we can choose the third vertex from the remaining vertices that are not adjacent to the chosen side. For each side of the polygon, the two vertices adjacent to it cannot be chosen.

For example, if we choose side AB, we cannot choose vertices A or B or their adjacent vertices C and J. Therefore, we can choose the third vertex from the remaining 6 vertices.

Since there are 10 sides in the polygon, the number of triangles that can be formed using one side of the polygon is:

\[ \text{Triangles using one side} = 10 \times 6 = 60 \]

#### b. Triangles using two sides of the polygon:

If we choose two sides of the polygon, we can only form one triangle. For example, if we take sides AB and BC, the only triangle we can form is ABC. There are 10 such triangles (one for each side).

So, the number of triangles that can be formed using two sides of the polygon is:

\[ \text{Triangles using two sides} = 10 \]

### Step 4: Calculate the total unwanted triangles.

Now, we can find the total number of unwanted triangles:

\[ \text{Total unwanted triangles} = \text{Triangles using one side} + \text{Triangles using two sides} = 60 + 10 = 70 \]

### Step 5: Subtract the unwanted triangles from the total triangles.

Finally, we subtract the unwanted triangles from the total triangles:

\[ \text{Valid triangles} = \text{Total triangles} - \text{Total unwanted triangles} = 120 - 70 = 50 \]

### Conclusion

Thus, the number of triangles that can be formed by joining the vertices of the polygon, which do not share any side with the polygon, is **50**.

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