Schrodinger's Lamps!

Let kk and nn be positive integers with k=n+10k=n+10. Let 2n2n lamps labelled 1,2,3...,2n1,2,3...,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 NN be the number of such sequences consisting of kk steps and resulting in the state where lamps 11 through nn are all on, and lamps n+1n+1 through 2n2n are all off.

Let MM be number of such sequences consisting of kk steps, resulting in the state where lamps 11 through nn are all on, and lamps n+1n+1 through 2n2n are all off, but where none of the lamps n+1n+1 through 2n2n is ever switched on.

Find NM.\frac{N}{M}.

×

Problem Loading...

Note Loading...

Set Loading...