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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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