2023 AIME II Problem 11

Find the number of collections of $16$ distinct subsets of $\{1, 2, 3, 4, 5\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X\cap Y \neq \emptyset$.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.