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📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 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.

📌 Favorable: assignments where all urns have at least one marble.

📌 Subtract assignments missing at least one urn.

📌 Thus, the correct model is 81 outcomes.

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