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Discrete Mathematics Level pending

$$10$$ coins are flipped randomly by Bob, and then placed in a line.

For an arrangement of coins $$S$$, suppose $$s$$ is a subset of consecutive elements of $$S$$ (meaning that they appear next to each other in the line). Let $$U(s)$$ and $$D(s)$$ be the number of coins facing up and down in $$s$$, respectively. $$S$$ is considered to be $$n-close$$ if for any $$s$$, $$\left| U(s)-D(s) \right| \le n$$.

Given that an arrangement $$S$$ is $$3-close$$, what is the probability that it is also $$2-close$$? Express your answer as $$\frac { m }{ n }$$ where $$m$$ and $$n$$ are relatively prime natural numbers. What is $$m+n$$?

Tip: There's a nice, simple way to solve this without casework. Don't make things too complicated!

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