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Multiply both sides by 5 to eliminate the fraction: - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Each crevasse can be labeled using one of the 3 symbols: A, B, or C. Since the symbols can repeat, the number of ways to label each crevasse is 3. Given there are 5 crevasses, and each can be independently labeled, the total number of distinct labeling arrangements is calculated by raising the number of choices per crevasse to the power of the number of crevasses:

📌 Thus, the total number of distinct labeling arrangements is \(\boxed{243}\).

📌 A historian is cataloging 6 unique historical documents into 2 indistinguishable archival boxes. In how many distinct ways can this be done?

📌 We need to partition the 6 unique documents into 2 indistinguishable boxes. This is equivalent to finding the number of distinct partitions of a set of 6 elements into 2 non-empty subsets. The number of ways to divide \(n\) distinct items into 2 non-empty indistinguishable subsets is given by:

📌 \frac{1}{2} \left(2^n - 2\right)

📌 Here, \(n = 6\):

📌 \frac{1}{2} \left(2^6 - 2\right) = \frac{1}{2} (64 - 2) = \frac{62}{2} = 31

📌 The subtraction of 2 accounts for excluding the cases where all items are in one box (leaving the other empty), and division by 2 adjusts for indistinguishability (since swapping the two boxes yields the same configuration).

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