Triangles are formed by joining vertices of a octagon then numbr of triangle

Answer

To find the number of triangles that can be formed by joining the vertices of an octagon, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Number of Vertices**:
An octagon has 8 vertices. Let's denote these vertices as \( A_1, A_2, A_3, A_4, A_5, A_6, A_7, A_8 \).

2. **Understand Triangle Formation**:
A triangle is formed by selecting any 3 vertices from these 8 vertices.

3. **Use the Combination Formula**:
The number of ways to choose 3 vertices from 8 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, \( n = 8 \) and \( r = 3 \).

\[ \text{Number of triangles} = \binom{8}{3} \]

4. **Calculate the Combination**:
The formula for combinations is:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Applying this to our case:

\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} \]

5. **Simplify the Factorial Expression**:
We can simplify \( \frac{8!}{3! \cdot 5!} \):

\[ \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} \]

The \( 5! \) in the numerator and denominator cancels out:

\[ = \frac{8 \times 7 \times 6}{3!} \]

6. **Calculate \( 3! \)**:
\( 3! = 3 \times 2 \times 1 = 6 \).

7. **Final Calculation**:
Now substitute \( 3! \) back into the equation:

\[ = \frac{8 \times 7 \times 6}{6} \]

The \( 6 \) cancels out:

\[ = 8 \times 7 = 56 \]

8. **Conclusion**:
Therefore, the total number of triangles that can be formed by joining the vertices of an octagon is **56**.