Question 12
Given the numbers $1, 2, 2^{2}, \ldots, 2^{n-1}$. For a specific permutation $\sigma = X_{1}, X_{2}, \ldots, X_{n}$ of these numbers we define $S_{1}(\sigma) = X_{1}$, $S_{2}(\sigma) = X_{1} + X_{2}$, $S_{3}(\sigma) = X_{1} + X_{2} + X_{3}$, $\ldots$ and $Q(\sigma) = S_{1}(\sigma) S_{2}(\sigma) \cdots S_{n}(\sigma)$. Evaluate $\sum 1 / Q(\sigma)$ where the sum is taken over all possible permutations.