Solution: Since the sensors are indistinct in physical position (the robots configuration depends only on how many sensors are in each mode, not their spatial arrangement), this is a problem of counting the number of non-negative integer solutions to the equation:

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**The Hidden Math Behind Robot Sensor Configurations — Why Counting Modes Matters More Than You Think** As automation and intelligent robotics grow faster than ever, a quiet challenge emerges at the heart of their design: how to define configuration without relying on precise spatial placement. This is especially true in systems where sensors are indistinct in physical positioning — their role defined not by where they sit, but by how many are active in each operational mode. The core question often surfaces in technical discussions: *How many unique ways can sensors be distributed across modes when their physical orientation doesn’t influence configuration?* The answer lies in a core principle of combinatorics — counting non-negative integer solutions to a simple equation — a problem gaining traction across engineering, e-commerce, and even emerging AI robotics platforms. This isn’t just academic curiosity. As industries increasingly standardize sensor-based systems—from warehouse automation to smart home devices—understanding how to model configuration flexibility becomes a competitive and practical necessity. The challenge centers on finding the number of ways to assign sensors to operational modes, where each mode represents a defined function and repetition is allowed across units. This mathematical model reveals hidden variability in scalable robot setups, offering a clearer way to evaluate performance, upgrade paths, and maintenance budgets. **Why the Math Around Sensor Modes Is Intrigue-Driving Now**