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📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 5Question: A soil scientist at the USDA is testing 6 different organic compost mixtures across 4 identical experimental plots. If each plot must receive at least one mixture and mixtures are distinguishable but plots are not, how many distinct ways can the mixtures be assigned?

📌 Solution: We are distributing 6 distinguishable compost mixtures into 4 indistinguishable plots, with no plot left empty. This is equivalent to finding the number of ways to partition a set of 6 labeled elements into exactly 4 non-empty unlabeled subsets. This is given by the Stirling numbers of the second kind, denoted $ S(n, k) $. We seek $ S(6, 4) $.

📌 Question: An ornithologist tracks 7 distinct migratory birds using GPS tags and assigns each bird one of 3 predefined migration routes, where routes are treated as indistinct groupings (i.e., only the set of birds per route matters, not the route label). How many distinct grouping configurations are possible if each route must be used at least once?

📌 Solution: We are partitioning 7 distinguishable birds into exactly 3 non-empty, indistinguishable subsets (migration routes treated as unlabeled). This is again a Stirling number of the second kind, $ S(7, 3) $.

📌 Question: A museum curator is arranging 8 rare astrolabes and 5 orreries around a circular display table. If all items are distinct and rotations of the arrangement are considered identical, in how many ways can they be arranged so that no two orreries are adjacent?

📌 Fix one astrolabe at a position. Then, we arrange the remaining 7 astrolabes and 5 orreries in the remaining 12 positions, with the constraint that no two orreries are adjacent.

📌 First, arrange the 7 remaining astrolabes in the remaining 12 positions. But better approach: place the astrolabes first to create safe gaps.

📌 Now, assign the 5 distinct orreries to these 5 selected gaps: $ 5! $ ways.

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