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I'm answering this hoping you know the basics of graph theory.

Going with these notations, $r =$degree of region, $v=$ degree of vertex and $e =$ degree of an edge.

Now,

1. Exactly 5 edges in boundary of each region : degree of a region is $5$. So $5r = 2e$
2. Cubic planar graph :  degree of every vertex is $3$. So $3v = 2e$.
3. Cubic planar graph : Using Euler's Formula, we have $r = e - v + 2$

Now we just need to solve for $e$, using the equations above.

$\frac{2}{5} \times e = e - \frac{2}{3} \times e + 2 \Rightarrow 6e = 15e - 10e + 30 \Rightarrow e = 30$

:) sudoankit