Let $G$ be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in $G$ is ____________.

Let:

• $V = 10$ be the number of vertices

• $E$ be the number of edges

• $F$ be the number of faces

Since each face is bounded by exactly $3$ edges and every edge is shared by $2$ faces (or counted twice across face boundaries):

Using Euler's formula for a connected planar graph:

Substitute $V = 10$ and $F = \frac{2E}{3}$:

Thus, the number of edges in $G$ is 24.