2022 AIME I Problems/Problem 12

.= Problem == 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 $\left\{ 1 , 2 , 3, \cdots , n \right\}$ with $|A| = |B|$ . For example, $S_2 = 4$ because the sum is taken over the pairs of subsets $(A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2\} ) , ( \{1 , 2\} , \{1 , 2\} ) \right\} ,$ 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.

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2022 AIME I Problems/Problem 12