# The Road Not Taken...

Suppose you have $$24$$ identical toothpicks and arrange them so that they form an outline of a $$3$$ by $$3$$ grid of $$9$$ identical squares.

Starting at the lower left corner, we can trace out $$20$$ distinct paths to the top right corner by going either up or to the right, one toothpick at a time. (Each of these paths involve precisely $$6$$ toothpicks.)

If we remove one of the $$24$$ toothpicks at random, what is the expected number of these $$20$$ distinct paths that will remain intact?

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