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Home » Not helpful unless complex conjugates known. - Databee Business Systems

Not helpful unless complex conjugates known. - Databee Business Systems

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Plugging in $ n = 7 $, $ k = 3 $:

📌 S(7, 3) = \frac{1}{6} \left( 3^7 - 3 \cdot 2^7 + 3 \cdot 1^7 \right) = \frac{1}{6} (2187 - 3 \cdot 128 + 3) = \frac{1}{6} (2187 - 384 + 3) = \frac{1806}{6} = 301

📌 Thus, the number of ways is $\boxed{301}$.

📌 Question: A software developer is training a neural network with 4 distinct image filters. How many ways can they assign these filters to 2 identical processing modules, allowing for empty modules?

📌 Solution: Since the modules are indistinguishable and empty modules are allowed, the number of ways is the sum of Stirling numbers of the second kind for partitions into 1 and 2 non-empty subsets:

📌 S(4, 1) + S(4, 2) = 1 + 7 = 8

📌 Here, $ S(4, 1) = 1 $ (all filters in one module) and $ S(4, 2) = 7 $ (ways to split 4 filters into 2 non-empty groups). Thus, the total is $\boxed{8}$.

📌 Question: An angel investor is evaluating 6 biotechnology startups. What is the probability that at least 2 of the 3 randomly selected startups focus on gene editing, given that 4 of the 6 startups specialize in this area?

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