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📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Solution: The list is simply the positive integers:

📌 The 50th number is $ \boxed{50} $.

📌 Question: A student measures two strands of synthetic fiber, one 3.75 cm long and the other 8.25 cm long. What is the average length, in centimeters?

📌 Solution: Compute the average:

📌 So, the average length is $ \boxed{6} $ centimeters.Question: An archaeologist is analyzing 7 distinct artifacts found at an Inca settlement, and wishes to divide them into 3 identical storage containers such that each container holds at least one artifact. How many distinct ways can this be done?

📌 Solution: We are to partition 7 distinguishable artifacts into 3 indistinguishable non-empty containers. This is equivalent to finding the Stirling numbers of the second kind, denoted $ S(n, k) $, which count the number of ways to partition $ n $ distinguishable objects into $ k $ non-empty indistinguishable subsets.

📌 We need $ S(7, 3) $. The value of $ S(7, 3) $ can be computed using the recurrence:

📌 S(n, k) = k \cdot S(n-1, k) + S(n-1, k-1),

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