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average steps in a mile - ClickBalance

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 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📌 A historian is cataloging 6 unique historical documents into 2 indistinguishable archival boxes.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).

📌 Thus, the number of distinct ways is \(\boxed{31}\).

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