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Home » Thus, the total volume is $ \boxed{\frac83 \pi r^3} $. - Silent Sales Machine

Thus, the total volume is $ \boxed{\frac83 \pi r^3} $. - Silent Sales Machine

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Treat R1 and R2 as a single block. Then we have 4 units to permute: the block and the other 3 rituals.

📌 Number of ways: \(4! = 24\), and within the block, R1 and R2 can appear as (R1,R2) or (R2,R1): 2 ways.

📌 So total invalid arrangements: \(24 \times 2 = 48\)

📌 Thus, valid arrangements: \(120 - 48 = 72\)

📌 So the answer is \(\boxed{72}\)

📌 An electrical engineer designs a smart grid system with 8 distinct subsystems, each capable of being in one of two states: active or inactive. How many configurations exist such that exactly 3 subsystems are active, and no two active subsystems are adjacent?

📌 We model this as placing 3 active (A) subsystems and 5 inactive (I) subsystems in a line of 8 positions, such that no two A’s are adjacent.

📌 To ensure no two A’s are adjacent, first place the 5 inactive subsystems:

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