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\boxed(1, 2, 3) - Londonproperty

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 Question: A computational biologist studies a genome with 8 genes. How many ways can 3 genes be selected for further analysis if the order of selection does not matter?

📌 Solution: The number of ways to choose 3 genes from 8 is given by the combination formula:

📌 Thus, there are $ \boxed{56} $ possible selections.

📌 Question: A Mars colony’s AI schedules two solar panel maintenance tasks at random times between 10:00 and 11:00. What is the probability that the second task starts at least 20 minutes after the first?

📌 Solution: Represent the times as $(x, y)$ in a 60-minute square. The condition $ y \geq x + 20 $ defines a triangular region with base $ 60 - 20 = 40 $ minutes. The area of this region is $ \frac{1}{2} \times 40^2 = 800 $. The total area is $ 60^2 = 3600 $. The probability is $ \frac{800}{3600} = \frac{2}{9} $.

📌 Thus, the probability is $ \boxed{\dfrac{2}{9}} $.

📌 Question: A piezoelectric road wear engineer needs to distribute 4 distinct sensors into 2 identical sections. How many ways can this be done if each section must contain at least one sensor?

📌 Solution: This is equivalent to partitioning 4 distinguishable objects into 2 non-empty indistinct subsets, calculated by the Stirling number $ S(4, 2) $. Using the recurrence:

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