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california state polytechnic university-pomona - Deep Underground Poetry

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Valid communication patterns:

📌 Thus, the number of distinct communication patterns with at least two gestures is $ \boxed{120} $.

📌 Question: A solar energy technician must install 6 identical photovoltaic panels on 4 distinguishable rooftops, such that each rooftop gets at least one panel. How many valid installation configurations are there?

📌 Solution: We are distributing 6 identical panels into 4 distinguishable rooftops with each rooftop receiving at least one panel. This is a classic stars and bars problem with the constraint of positive integers.

📌 Let $ x_1 + x_2 + x_3 + x_4 = 6 $, where $ x_i \geq 1 $.

📌 Substitute $ y_i = x_i - 1 $, so $ y_i \geq 0 $ and:

📌 y_1 + y_2 + y_3 + y_4 = 6 - 4 = 2

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