Question 12
For any finite set $X,$ let $|X|$ denote the number of elements in $X.$ Define $$
S_n = \sum |A \cap B|,
$$ where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\{1, 2, 3, \ldots, n\}$ with $|A| = |B|.$ For example, $S_2 = 4$ because the sum is taken over the pairs of subsets $$
\begin{aligned} &(A, B) \in {}\\ &\quad \small\left\{\begin{gathered} (\emptyset, \emptyset), (\{1\}, \{1\}), \\ (\{1\}, \{2\}), (\{2\}, \{1\}), \\ (\{2\}, \{2\}), (\{1, 2\}, \{1, 2\}) \end{gathered}\right\}, \end{aligned}
$$ giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4.$ Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by $1000.$
S_n = \sum |A \cap B|,
$$ where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\{1, 2, 3, \ldots, n\}$ with $|A| = |B|.$ For example, $S_2 = 4$ because the sum is taken over the pairs of subsets $$
\begin{aligned} &(A, B) \in {}\\ &\quad \small\left\{\begin{gathered} (\emptyset, \emptyset), (\{1\}, \{1\}), \\ (\{1\}, \{2\}), (\{2\}, \{1\}), \\ (\{2\}, \{2\}), (\{1, 2\}, \{1, 2\}) \end{gathered}\right\}, \end{aligned}
$$ giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4.$ Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by $1000.$