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non renewable resources and renewable resources examples - Anadea

📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 Question: A science educator is developing a 6-day interactive virtual lab path, where each dayâs activity involves one of 4 experiment types: Physics, Chemistry, Biology, or Astronomy. If each experiment type must be used at least once during the week and activities are distinguishable only by type, how many distinct 6-day sequences can be designed?

📌 Solution: We are counting the number of 6-day sequences using 4 experiment types (P, C, B, A), where each type appears at least once, and sequences are ordered. This is a classic inclusion-exclusion problem: counting the number of surjective functions from a 6-day sequence to 4 experiment types.

📌 Total sequences without restriction: $ 4^6 $

📌 Subtract sequences missing at least one type:

📌 Let $ S = 4^6 $

📌 Use inclusion-exclusion:

📌 ext{Number of surjective sequences} = \sum_{k=0}^{4} (-1)^k inom{4}{k} (4 - k)^6

📌 $ k = 0 $: $ (-1)^0 inom{4}{0} \cdot 4^6 = 1 \cdot 1 \cdot 4096 = 4096 $

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