# On tap tonight...

Suppose that we have a row of $$5$$ taps, all of which run into the same tank. Each tap fills the tank at a different rate, such that the $$n$$th tap can fill the tank in $$n$$ hours.

Two taps, chosen at random, are turned on and allowed to fill the tank. The expected time, in hours, that it will take to fill the tank is $$\dfrac{a}{b}$$, where $$a$$ and $$b$$ are positive coprime integers. Find $$a - b$$.

Note: You may need a calculator for the last step of the problem.

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