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📅 July 7, 2026 👤 scraface

Jul 07, 2026

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📌 Number of circular arrangements of 4 entities:

📌 Thus, the probability that all computer scientists sit together is:

📌 A home-schooled student is simulating a game where four indistinguishable marbles are placed into three distinguishable urns labeled A, B, and C. What is the probability that no urn is empty, given that each marble independently lands in one of the urns with equal probability?

📌 We compute the probability that no urn is empty when 4 indistinguishable marbles are randomly distributed into 3 distinguishable urns (each marble chooses an urn independently and uniformly).

📌 But since marbles are indistinguishable and urns are distinguishable, we model this by counting integer solutions to:

📌 Total number of non-negative integer solutions (stars and bars):

📌 Alternatively, list them: partitions of 4 into 3 positive integers:

📌 No other: (1,1,1,1) invalid, (3,1,1) sums to 5. So only (2,1,1) and permutations.

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