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$ f(x) = 0 $ for all $ x $ - SUpost

📅 July 5, 2026 👤 scraface

Jul 05, 2026

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📌 Question: A hydrologist is modeling water flow through 6 distinct layers of soil, each with unique permeability. In how many ways can the layers be arranged vertically so that the most permeable layer is not directly above the least permeable layer?

📌 Solution: First, compute total number of permutations of the 6 distinct layers:

📌 Now count the number of arrangements where the most permeable layer is directly above the least permeable layer. Treat the pair (least, most) as a single unit. There are 5 positions this pair can occupy in a vertical stack of 6: positions 1–2, 2–3, ..., 5–6. For each such position, the pair is fixed in order, and the remaining 4 layers can be arranged in:

📌 So total number of invalid arrangements is:

📌 \times 24 = 120

📌 Thus, the number of valid arrangements is:

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