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).
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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.
- 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 \(n\).
