The vertices and faces of a polyhedron are 32 and 15, respectively. Find the number of edges of this polyhedron.

Options

A. 44
B. 46
C. 45
D. 47

Answer

To find the number of edges of the polyhedron given the number of vertices and faces, we can use Euler's formula for polyhedra, which states:

\[ V - E + F = 2 \]

where:
- \( V \) is the number of vertices,
- \( E \) is the number of edges,
- \( F \) is the number of faces.

Given:
- \( V = 32 \) (the number of vertices)
- \( F = 15 \) (the number of faces)

We need to find \( E \) (the number of edges).

### Step-by-step Solution:

1. **Write down Euler's formula**:
\[ V - E + F = 2 \]

2. **Substitute the known values into the formula**:
\[ 32 - E + 15 = 2 \]

3. **Combine the constants on the left side**:
\[ 47 - E = 2 \]

4. **Isolate \( E \)**:
\[ -E = 2 - 47 \]
\[ -E = -45 \]
\[ E = 45 \]

Thus, the number of edges \( E \) of the polyhedron is **45**.