OME 2021 Problem 5

We have $2n$ lights in two rows, numbered from $1$ to $n$ in each row. Some (or none) of the lights are on and the others are off, we call that a "state". Two states are distinct if there is a light which is on in one of them and off in the other. We say that a state is good if there is the same number of lights turned on in the first row and in the second row.

Prove that the total number of good states divided by the total number of states is: $$\frac{3 \cdot 5 \cdot 7 \cdots (2n-1)}{2^n n!}$$