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 1: Identify the number of vertices in the octagon
An octagon has 8 vertices.

### Step 2: Understand how triangles are formed
A triangle is formed by selecting 3 vertices. Therefore, we need to choose 3 vertices from the 8 available vertices of the octagon.

### Step 3: Use the combination formula
The number of ways to choose 3 vertices from 8 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
In this case, \( n = 8 \) and \( r = 3 \).

### Step 4: Substitute the values into the formula
Substituting the values into the formula gives us:
\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} \]

### Step 5: Simplify the factorials
Now, we can simplify the factorials:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} \]
The \( 5! \) in the numerator and denominator cancels out:
\[ = \frac{8 \times 7 \times 6}{3!} \]

### Step 6: Calculate \( 3! \)
Now calculate \( 3! \):
\[ 3! = 3 \times 2 \times 1 = 6 \]

### Step 7: Substitute back and calculate
Now substitute \( 3! \) back into the equation:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{6} \]
The \( 6 \) in the numerator and denominator cancels out:
\[ = 8 \times 7 = 56 \]

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

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