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📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 We are distributing 7 distinguishable coral colonies into 2 indistinguishable zones, with no zone empty. Since the zones are identical, the order does not matter, and we are counting the number of **partitions of a 7-element set into exactly 2 non-empty unlabeled subsets**.

📌 This is given by the Stirling number of the second kind \(S(7,2)\), which counts the number of ways to partition 7 labeled objects into 2 non-empty unlabeled subsets.

📌 The formula for \(S(n,2)\) is:

📌 S(n,2) = 2^{n-1} - 1

📌 For \(n = 7\):

📌 S(7,2) = 2^{6} - 1 = 64 - 1 = 63

📌 Alternatively, each object can go into one of the two subsets, giving \(2^7 = 128\) labeled assignments, but dividing by 2 for indistinguishability gives \(64\), and subtracting the two cases where all are in one zone (leaving the other empty), we get \(64 - 1 = 63\) (we subtract 1 for the all-empty-one-case, but actually, each symmetric split is counted once, so the standard result applies directly).

📌 Thus, the number of distinct groupings is \(\boxed{63}\).

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