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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 \binom{3}{2} \times (2^5 - 1) = 3 \times 31 = 93

📌 Thus, the number of combinations is \(\boxed{93}\).

📌 Question:** A marine biologist in the Caribbean has tagged 7 distinct coral colonies and wants to place them into 2 identical monitoring zones, with no zone left empty. How many distinct groupings are possible?

📌 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 \(n = 7\):

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