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Home » Number of arrangements: $ \frac4!2!1!1! = 12 $ - Portal da AcÃºstica

Number of arrangements: $ \frac4!2!1!1! = 12 $ - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 An electrical engineer is configuring 5 distinct smart meters to be installed at different substations arranged in a line. Each meter is assigned one of two power modes: high-efficiency (H) or standard (S). How many assignments are there such that no two adjacent meters use high-efficiency mode?

📌 We are assigning a mode (H or S) to each of 5 substations in a line, with 5 distinct positions (so labels matter), but the constraint is only on adjacency of H’s.

📌 Let’s first count the number of valid H/S assignments (ignoring position labels for structure, then multiply by permutations? No — since the substations are distinct and fixed in line, each assignment corresponds to a binary string of length 5 with no two consecutive H’s.

📌 But the "positions" are fixed and distinct, so we are counting binary strings of length 5 with no two consecutive 1’s (let H = 1, S = 0).

📌 This is a classic problem: number of binary strings of length \(n\) with no two consecutive 1’s is \(F_{n+2}\), where \(F_n\) is the Fibonacci sequence.

📌 Standard recurrence:

📌 Let \(a_n\) = number of such strings of length \(n\).

📌 \(a_1 = 2\) (0, 1),

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