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\boxed{\dfrac932} - DevRocket

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Similarly, \(A \cap C = 1\), \(B \cap C = 1\),

📌 \(A \cap B \cap C = 0\) (impossible to have no types).

📌 By inclusion-exclusion, the number of invalid configurations (missing at least one type) is:

📌 |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|

📌 = 3 \cdot 512 - 3 \cdot 1 + 0 = 1536 - 3 = 1533

📌 Thus, valid configurations with all three types present:

📌 3^9 - 1533 = 19683 - 1533 = 18150

📌 Therefore, the number of valid configurations is \(\boxed{18150}\).

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