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In a connected planar graph, every vertex has degree $3$, and every face is bordered by $5$ or $6$ edges. How many faces are bordered by $5$ edges?

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Plug everything in the Euler's formula: $v-e+f=2$

$v=$#of vertices

$e=$#of edges

$f=$#of faces

$p=$#of pentagonal faces

$h=$#of hexagonal faces

So we have:

$e=\frac{5p+6h}{2}$ (each edge belongs exactly to two faces)

$v=\frac{5p+6h}{3}$ (each vertex is shared by exactly three faces)

$f=p+h$ (each face is either pentagonal or hexagonal)

Plugging into Euler's equation gives:

$\frac{5p+6h}{3}-\frac{5p+6h}{2}+(p+h)=2$

$h$ cancels out and the solution is $p=12$

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