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Home » From Backyard Pets to Pokémon Champions: The Dog Pokémon Revolution You Created! - Appcentric

From Backyard Pets to Pokémon Champions: The Dog Pokémon Revolution You Created! - Appcentric

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 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.

📌 But if the "order" refers only to the application of the operation (i.e., 2 positions are distinct for pumping), the count is $ \binom{7}{4} \cdot \binom{4}{2} \cdot \frac{4!}{2!} = 35 \cdot 6 \cdot 12 = 2520 $. Clarifying the problem's intent, the most plausible interpretation leads to $\boxed{2520}$.

📌 Question: A nanotechnology engineer is designing a solar-powered water harvesting system with 5 identical microfilters. How many distinct ways can 8 unique water source samples be distributed into the filters if each filter can hold any number of samples?

📌 Solution: Since the filters are identical and samples are distinguishable, this is equivalent to partitioning 8 distinct elements into up to 5 indistinct subsets. The total number of ways is the sum of Stirling numbers of the second kind for $ k = 1 $ to $ 5 $:

📌 S(8, 1) + S(8, 2) + S(8, 3) + S(8, 4) + S(8, 5).

📌 Using known values:

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