Schrodinger's Lamps!

Let \(k\) and \(n\) be positive integers with \(k=n+10\). Let \(2n\) lamps labelled \(1,2,3...,2n\) be given, each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).

Let \(N\) be the number of such sequences consisting of \(k\) steps and resulting in the state where lamps \(1\) through \(n\) are all on, and lamps \(n+1\) through \(2n\) are all off.

Let \(M\) be number of such sequences consisting of \(k\) steps, resulting in the state where lamps \(1\) through \(n\) are all on, and lamps \(n+1\) through \(2n\) are all off, but where none of the lamps \(n+1\) through \(2n\) is ever switched on.

Find \(\frac{N}{M}.\)

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