Find the number of diagonals in an n-sided polygon.

Options

A. `n^2`
B. `(n(n-3))/(2)`
C. `n!`
D. `2^n`

Answer

To find the number of diagonals in an n-sided polygon, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Polygon**:
- An n-sided polygon has n vertices.

2. **Finding Line Segments**:
- The total number of line segments that can be formed by connecting any two vertices of the polygon is given by the combination formula \( \binom{n}{2} \), which represents the number of ways to choose 2 vertices from n vertices.
- This can be calculated as:
\[ \binom{n}{2} = \frac{n!}{2!(n-2)!} = \frac{n(n-1)}{2} \]

3. **Counting the Sides of the Polygon**:
- An n-sided polygon has exactly n sides.

4. **Calculating the Number of Diagonals**:
- The diagonals of the polygon are the line segments that connect two non-adjacent vertices. Therefore, to find the number of diagonals, we subtract the number of sides from the total number of line segments:
\[ \text{Number of Diagonals} = \text{Total Line Segments} - \text{Number of Sides} \]
- Substituting the values we have:
\[ \text{Number of Diagonals} = \frac{n(n-1)}{2} - n \]

5. **Simplifying the Expression**:
- To simplify \( \frac{n(n-1)}{2} - n \):
\[ = \frac{n(n-1)}{2} - \frac{2n}{2} = \frac{n(n-1) - 2n}{2} = \frac{n^2 - n - 2n}{2} = \frac{n^2 - 3n}{2} \]
- This can be factored as:
\[ = \frac{n(n-3)}{2} \]

6. **Final Result**:
- Therefore, the number of diagonals in an n-sided polygon is:
\[ \text{Number of Diagonals} = \frac{n(n-3)}{2} \]