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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