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📅 May 10, 2026 👤 scraface

May 10, 2026

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📌 At \( t = 4 \), \( a(4) = 6(4) + 2 = 24 + 2 = 26 \) m/s².

📌 #### 26 m/s²Question: A climate policy analyst is evaluating the distribution of 6 distinct urban green energy initiatives into 3 identical community districts. How many distinct ways can this be done if each district must receive at least one initiative?

📌 Solution: The problem requires partitioning 6 distinguishable initiatives into 3 non-empty indistinct subsets. This is given by the Stirling numbers of the second kind, $ S(6, 3) $. The formula for $ S(n, k) $ is $ \frac{1}{k!} \sum_{i=0}^{k} (-1)^{k-i} \binom{k}{i} i^n} $. Plugging in $ n=6 $ and $ k=3 $:

📌 S(6, 3) = \frac{1}{6} \left( 3^6 - 3 \cdot 2^6 + 3 \cdot 1^6 \right) = \frac{1}{6} (729 - 192 + 3) = \frac{540}{6} = 90.

📌 Thus, the number of ways is $\boxed{90}$.

📌 Question: A hydrologist is modeling groundwater flow through 7 distinct soil layers. How many distinct sequences of 4 layers can be selected and ordered for analysis if the operation "pump water" is applied to exactly 2 of them?

📌 Solution: First, choose 4 distinct layers from 7, which is $ \binom{7}{4} = 35 $. Then, select 2 of these 4 layers to apply the "pump water" operation, which can be done in $ \binom{4}{2} = 6 $ ways. The remaining 2 layers are analyzed without the operation, and the sequence of 4 layers is ordered, giving $ 4! = 24 $ permutations. However, since the operation affects only the selection (not the order of non-pump layers), the total is:

📌 \binom{7}{4} \cdot \binom{4}{2} \cdot 4! = 35 \cdot 6 \cdot 24 = 5040.

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