Question 133
Let $A = \{a_1, a_2, \dots, a_n\}$ be a set of positive integers with $n \ge 1$ elements. The set $A$ is called good if, for any two distinct subsets $X$ and $Y$ of $A$ (i.e., $X \neq Y$), the difference $S(X) - S(Y)$ is not divisible by $2^n$. Here, $S(X) = \sum_{a \in X} a$ denotes the sum of the elements in the subset $X \subseteq A$, and $S(\emptyset) = 0$ by definition.
Determine the number of good sets $A$ such that all elements in $A$ are less than $2^n$.
(Bayarmagnai Gombodorj)
Determine the number of good sets $A$ such that all elements in $A$ are less than $2^n$.
(Bayarmagnai Gombodorj)