# Expected Area inside Hexagon

Geometry Level 4

A regular hexagon with side length $$1$$ is drawn. We flip $$6$$ coins, with each coin corresponding to a unique midpoint of the hexagon. Then, we take all the midpoints of which their corresponding coin resulted heads, and connect them in clockwise order to create a polygon inside the hexagon. If the expected value of the area of this polygon can be expressed as $\dfrac{a\sqrt{b}}{c}$ where $$a,b,c$$ are positive integers, $$a,c$$ are coprime and $$b$$ is square-free, then find $$a+b+c$$.

$$\text{Details and Assumptions}$$

A polygon with zero, one or two vertices has an area of $$0$$.

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