Question 53
Let $o(n)$ be the number of $2n$-tuples $(a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n})$ such that each $a_{i}, b_{j} = 0$ or $1$ and $a_{1} b_{1} + a_{2} b_{2} + \ldots + a_{n} b_{n}$ is odd. Similarly, let $e(n)$ be the number for which the sum is even. Show that $\dfrac{o(n)}{e(n)} = \dfrac{2^{n} - 1}{2^{n} + 1}$.