# a simple connected planar graph G with 10 vertices and 25 edges have 17 faces

Why cannot a simple connected planar graph G with 10 vertices and 25 edges have 17 faces?

by using Euler’s formula, $$n−m+f = 10 -25 +17=2$$.

How can I show that it does not?

You probably learned that for a simple connected planar graph,

$$\# \, edges\leq 3(\#\,vertices)-6$$

Which shows that your graph is impossible, since the following inequality is clearly false: $$25 \leq 3(10)-6 = 24$$ Otherwise, a proof of the above inequality can be found on the internet as a consequence of Euler's Formula (or from another answerer on this post, or from me this afternoon)