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📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 Question: A researcher studying microfossil assemblages collects samples from 5 stratigraphic layers, each containing fossils from 4 different geological epochs. If one fossil is selected from each layer, and no two adjacent layers can contribute a fossil from the same epoch, how many valid selections are possible?

📌 Solution: We are to count the number of sequences of length 5 (one per layer), where each element is an epoch from 1 to 4, and no two consecutive layers have the same epoch.

📌 This is a classic recurrence problem.

📌 Let $ a_n $ be the number of valid sequences of length $ n $ with 4 choices per position, no two adjacent the same.

📌 For the first layer: 4 choices.

📌 For each subsequent layer: 3 choices (different from the previous).

📌 So total: $ a_5 = 4 \cdot 3^4 $

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