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#### 140 degrés - go-checkin.com

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Solution: The scalar triple product equals the volume of the parallelepiped. Maximum occurs when vectors are orthogonal, giving $|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| = 1 \cdot 1 \cdot 1 \cdot \sin(90^\circ)\cos(0^\circ) = 1$.

📌 \boxed{1}Question: A research scientist is preparing 7 different chemical solutions. How many ways can they distribute these to 3 identical storage vials, ensuring each vial contains at least one solution?

📌 Solution: The problem requires partitioning 7 distinguishable solutions into 3 indistinguishable non-empty vials. This is equivalent to the Stirling numbers of the second kind, $ S(7, 3) $. The formula for Stirling numbers is:

📌 S(n, k) = \frac{1}{k!} \sum_{i=0}^k (-1)^{k-i} \binom{k}{i} i^n}

📌 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?

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