Question 138
Let $S$ be the set of all $n$ tuples $(X_{1}, \ldots, X_{n})$, where $X_{1}, \ldots, X_{n}$ are subsets of $\{1,2, \ldots, 1000\}$, which do not all have to be different and which can also be empty. For $a=(X_{1}, \ldots, X_{n}) \in S$ designate
$$
E(a)=\text{Number of elements from} X_{1} \cup \ldots \cup X_{n}
$$
Find an explicit expression for the sum
$$
\sum_{a \in S} E(a)
$$
$$
E(a)=\text{Number of elements from} X_{1} \cup \ldots \cup X_{n}
$$
Find an explicit expression for the sum
$$
\sum_{a \in S} E(a)
$$