Can a polyhedron have 10 faces, 20 edges and 15 vertices?

Options

A. Yes
B. No
C. Cannot be determined
D. None of the above

Explanation

To determine if a polyhedron can have 10 faces, 20 edges, and 15 vertices, we can use Euler's formula for polyhedra. Euler's formula states that:

\[ F + V - E = 2 \]

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

### Step-by-Step Solution:

1. **Identify the values**:
- Number of faces \( F = 10 \)
- Number of vertices \( V = 15 \)
- Number of edges \( E = 20 \)

2. **Substitute the values into Euler's formula**:
\[ F + V - E = 2 \]
Substituting the values we have:
\[ 10 + 15 - 20 = 2 \]

3. **Calculate the left side**:
- First, add the number of faces and vertices:
\[ 10 + 15 = 25 \]
- Then subtract the number of edges:
\[ 25 - 20 = 5 \]

4. **Compare the result with 2**:
- We find that:
\[ 5 \neq 2 \]

5. **Conclusion**:
Since the left side of Euler's formula does not equal 2, we conclude that it is not possible for a polyhedron to have 10 faces, 20 edges, and 15 vertices.

### Final Answer:
No, a polyhedron cannot have 10 faces, 20 edges, and 15 vertices.