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Home » 2r + 2 = (3 + \sqrt5)r - (3 + \sqrt5) - 500apps

2r + 2 = (3 + \sqrt5)r - (3 + \sqrt5) - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 \frac{10!}{5! \cdot 3! \cdot 2!}

📌 10! = 3628800,\quad 5! = 120,\quad 3! = 6,\quad 2! = 2

📌 \frac{3628800}{120 \cdot 6 \cdot 2} = \frac{3628800}{1440} = 2520

📌 Thus, the number of distinct activation orders is $\boxed{2520}$.

📌 \boxed{2520}Question: An industrial designer creates a series of modular shelves where each shelf has 4 more panels than the previous one. If the first shelf has 7 panels, how many shelves can be built with a total of 150 panels?

📌 Solution: Let the number of shelves be \( n \). The number of panels forms an arithmetic sequence: 7, 11, 15, ..., with first term \( a = 7 \) and common difference \( d = 4 \). The sum of the first \( n \) terms is:

📌 S_n = \frac{n}{2} \left(2a + (n - 1)d\right) = \frac{n}{2} \left(2(7) + (n - 1)(4)\right) = \frac{n}{2} (14 + 4n - 4) = \frac{n}{2} (4n + 10)

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