# Schrodinger's Lamps!

Discrete Mathematics Level 5

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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