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📅 May 19, 2026 👤 scraface

May 19, 2026

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📌 |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 \(\boxed{18150}\).

📌 A historian is organizing 4 rare manuscripts into 3 indistinguishable role-based archives: Royal, Scholarly, and Commercial. Each manuscript must be assigned to exactly one archive. How many distinct ways can the manuscripts be assigned, considering only the number per archive matters?

📌 We are assigning 4 distinguishable manuscripts to 3 indistinguishable archives, where only the counts per group matter (i.e., the assignment is determined by a partition of 4 into at most 3 parts, and the manuscripts are labeled, so we count labeled partitions into unlabeled boxes).

📌 We list all integer partitions of 4 into at most 3 parts, where order of parts does not matter (since archives are indistinguishable), and then for each, compute the number of ways to assign labeled manuscripts accordingly.

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