Find the number of vertices in a polyhedron which has 30 edges and 12 faces

Answer

To find the number of vertices in a polyhedron with 30 edges and 12 faces, we can use Euler's formula, which states:

\[ V + F = E + 2 \]

where:
- \( V \) = number of vertices
- \( F \) = number of faces
- \( E \) = number of edges

### Step-by-Step Solution:

1. **Identify the given values**:
- Number of edges \( E = 30 \)
- Number of faces \( F = 12 \)

2. **Substitute the values into Euler's formula**:
\[ V + F = E + 2 \]
Plugging in the values we have:
\[ V + 12 = 30 + 2 \]

3. **Calculate the right side of the equation**:
\[ 30 + 2 = 32 \]
So now the equation is:
\[ V + 12 = 32 \]

4. **Isolate \( V \)**:
To find \( V \), subtract 12 from both sides:
\[ V = 32 - 12 \]

5. **Perform the calculation**:
\[ V = 20 \]

6. **Conclusion**:
The number of vertices \( V \) in the polyhedron is 20.

### Summary:
The polyhedron has **20 vertices**.