Find amount of edges and vertices of a plane graph

Question

A plane graph $G$ has 7 faces: 3 triangles, 3 quadrilaterals and 1 pentagon. How many edges and vertices does it have?

How this task should be solved? I came up with nothing better, than just manually drawing it and counting edges/vertices. From here, of course, I have no chance to prove that there is no other solution.

Answer 1

Using @hardmath hint in comments, the solution is very straightforward:

1. 3 triangles, 3 quadrilaterals and 1 pentagon gives us 26 sides. Each side must be some edge. Each edge must belong precisely to 2 faces. So we have $26/2 = 13$ edges.

2. Using Euler's formula we see that $v + 7 - 13 = 2 \Rightarrow v = 8$