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2023 SSMO Team Round Problems/Problem 7

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Problem

Let $S = \{1, 2, 3, 4, \cdots, 23\}$ and let there be randomly chosen sets $A, B, C$ where $A, B, C \subseteq S$ . The probability that $|A| + |B| = |C|$ can be expressed as $\frac{m}{n}$ . Let $2^a$ be the largest power of $2$ such $2^a \mid n$ . Find $a$ .

Solution

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