<p>For any finite set <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" />, let <img alt="$|S|$" height="18" src="https://latex.artofproblemsolving.com/1/1/c/11ceba9d7455309f60178fe82bb6e81e6c01fa83.png" width="20" /> denote the number of elements in <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" />. Find the number of ordered pairs <img alt="$(A,B)$" height="18" src="https://latex.artofproblemsolving.com/b/b/0/bb0c75d9abe3e693ba0bb677cecbb923cefd81af.png" width="49" /> such that <img alt="$A$" height="14" src="https://latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png" width="15" /> and <img alt="$B$" height="14" src="https://latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png" width="17" /> are (not necessarily distinct) subsets of <img alt="$\{1,2,3,4,5\}$" height="20" src="https://latex.artofproblemsolving.com/d/2/2/d2289231163b75dca0daf0f842ee0fba6f289fc2.png" width="98" /> that satisfy<img alt="\[|A| \cdot |B| = |A \cap B| \cdot |A \cup B|\]" height="18" src="https://latex.artofproblemsolving.com/a/8/c/a8c5056f13bbfd71412164ca53b71c92395e5812.png" width="212" /></p> <p> </p>

Question No: 441

For any finite set $S$ , let $|S|$ denote the number of elements in $S$ . Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy $|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$

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Tags: Combinatorics

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