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Home » 2; "This AP Stats Formula Sheet Got Students Scoring 5s—Break It Down Now!" - ToelettAPP

2; "This AP Stats Formula Sheet Got Students Scoring 5s—Break It Down Now!" - ToelettAPP

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 We are assigning 5 distinguishable sessions to 3 distinguishable environments (forest, ocean, silence), such that each environment is used at least once. Since the sessions are distinct, this is a **surjective function** from a 5-element set to a 3-element set, with labeled outputs.

📌 The number of such surjective functions is given by:

📌 3^5 - \binom{3}{1 \cdot 2^5} + \binom{3}{2 \cdot 1^5} = 3^5 - 3 \cdot 2^5 + 3 \cdot 1^5

📌 This is an application of inclusion-exclusion:

📌 = 243 - 3 \cdot 32 + 3 \cdot 1 = 243 - 96 + 3 = 150

📌 Alternatively, using Stirling numbers of the second kind:

📌 \(S(5,3)\) counts partitions into 3 non-empty subsets: \(S(5,3) = 25\),

📌 Then assign the 3 subsets to the 3 distinct environments: \(3! = 6\),

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