Prove that Petersen's graph is non-planar using Euler's formula Prove that Petersen's graph is non-planar using Euler's formula. 
I know that $n - m + f = 2$. But should I count $f$ and prove that the summation does not equal to two or solve to get $f =7$ and argue that it is impossible???
We know the Petersen graph has 15 edges and 10 vertices.  In a planar graph, V+F-E=2.  In Petersen, that would be 10+F-15 = 2, so it would have 7 faces in it's planar embedding. The minimal cycle in Petersen is 5, so it would need to be made from pentagons, hexagons, or larger.

7 pentagons = 35 edges or more.  Half that, rounded down = 17 edges. Each edge can only be used twice, and we've gone over.
Using your notation, we have that $n=10$, $m=15$.  Inspecting the graph, we see that (1) each edge is included in exactly two faces, and (2) each cycle has a length of 5 or greater.  Combining these two facts yields $5f \le 2m.$  Substituting this into Euler's formula, we obtain a contradiction.