How many triangles can be formed by joining the vertices of a hexagon?

Options

A. 20
B. 6
C. 10
D. 30

Explanation

To find out how many triangles can be formed by joining the vertices of a hexagon, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Number of Vertices in the Hexagon**:
A hexagon has 6 vertices.

2. **Determine the Number of Vertices Needed to Form a Triangle**:
To form a triangle, we need to choose 3 vertices.

3. **Use the Combination Formula**:
The number of ways to choose 3 vertices from 6 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from (in this case, vertices), and \( r \) is the number of items to choose.

4. **Substitute the Values into the Formula**:
Here, \( n = 6 \) and \( r = 3 \):
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3! \cdot 3!} \]

5. **Calculate Factorials**:
- \( 6! = 6 \times 5 \times 4 \times 3! \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

Now substituting these into the equation:
\[ \binom{6}{3} = \frac{6 \times 5 \times 4 \times 3!}{3! \times 3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} \]

6. **Simplify the Expression**:
\[ = \frac{120}{6} = 20 \]

7. **Conclusion**:
Therefore, the number of triangles that can be formed by joining the vertices of a hexagon is **20**.