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Home » #### \(x = \frac215, y = \frac65\) - RoadRUNNER Motorcycle Touring & Travel Magazine

#### \(x = \frac215, y = \frac65\) - RoadRUNNER Motorcycle Touring & Travel Magazine

📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 5Question: A soil scientist is analyzing 7 distinct soil samples and wishes to divide them into 3 non-empty groups for comparative analysis, where the groups are indistinguishable. How many ways can this be done?

📌 Solution: We are tasked with partitioning 7 distinguishable soil samples into 3 non-empty, indistinguishable groups. This is equivalent to computing the Stirling number of the second kind, $ S(7, 3) $, which counts the number of ways to partition 7 labeled objects into 3 non-empty unlabeled subsets.

📌 The recurrence relation for Stirling numbers of the second kind is:

📌 We compute $ S(7, 3) $ step by step:

📌 $ S(4,1) = 1 $, $ S(4,2) = 7 $, $ S(4,3) = 6 $, $ S(4,4) = 1 $

📌 Question: A science administrator must allocate 8 different research grants to 4 distinct laboratories, such that each laboratory receives at least one grant. How many valid allocations are possible?

📌 Solution: We are distributing 8 distinguishable grants to 4 distinguishable laboratories, with each lab receiving at least one grant. This is a problem of counting surjective (onto) functions from a set of 8 elements to a set of 4 elements.

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