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Home » \frac(3u+2) + (5u+7) + (4u+1)3 = 62 - Veritas Home Health

\frac(3u+2) + (5u+7) + (4u+1)3 = 62 - Veritas Home Health

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 A glaciologist is studying a glacier with 5 distinct crevasses, each needing to be labeled with a unique identifier using just 3 different symbols: A, B, and C. How many distinct labeling arrangements are possible if each crevasse must be labeled and symbols can repeat?

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📌 A historian is cataloging 6 unique historical documents into 2 indistinguishable archival boxes.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

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