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- \frac16 = \frac56 - Simpleprint

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 A robotics engineer builds a hierarchical robot system with one command unit, 4 sensor clusters (all same, but placed differently), and 3 task modules (distinct functions: navigation, analysis, repair). Each sensor cluster can be assigned to any of 4 mounting positions, but due to space constraints, only 2 clusters can be placed per unit. How many distinct robot setups can be created?

📌 First, choose 2 sensor clusters out of 4: C(4,2) = 6.

📌 For each pair, assign the 2 chosen clusters to 4 distinct mounting positions with no two in the same spot: permutation P(4,2) = 4 × 3 = 12.

📌 Command unit: fixed (1 choice).

📌 Task modules: 3 distinct modules, assign one to each of 3 unique functions — 3! = 6 ways.

📌 Total setups: 6 (sensor clusters) × 12 (placement) × 6 (task modules) = 6 × 12 × 6 = <<6*12*6=432>>432.

📌 #### 432**Question:** A linguist is studying the evolution of language through a model where the frequency of a certain linguistic feature in a population is given by the function \( f(t) = 3t^2 - 2t + 5 \). Find the time \( t \) at which the frequency is 20.

📌 Solution:** To find the time \( t \) when the frequency \( f(t) = 20 \), we set up the equation:

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