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Home » \binom31 \times \binom73 - High Altitude Science

\binom31 \times \binom73 - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Solution: This requires the Stirling numbers of the second kind $ S(5, 3) $, which count the ways to partition 5 distinguishable objects into 3 non-empty indistinct subsets. Using the recurrence relation:

📌 S(5, 3) = S(4, 2) + 3 \times S(4, 3) = 7 + 3 \times 6 = 25

📌 Thus, there are $ \boxed{25} $ ways.

📌 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} $.

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