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Home » Solve for $\cos \theta$: - inBeat

Solve for $\cos \theta$: - inBeat

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 To solve this problem, we need to distribute 4 different types of virtual plants into 3 rooms such that each room gets at least one type. This is a combinatorial problem involving the concept of surjective functions, where each room (bucket) receives at least one plant (object).

📌 Next, we use the principle of inclusion-exclusion to subtract the cases where one or more rooms are empty.

📌 Subtract cases where at least one room is empty. Choose 1 room to be empty (3 ways), and distribute the 4 plants into the remaining 2 rooms. Each plant has 2 choices, so:

📌 Add back cases where exactly two rooms are empty (since they were subtracted twice). Choose 2 rooms to be empty (3 ways), and put all plants in the remaining room (1 way):

📌 Using inclusion-exclusion, the total number of valid distributions is:

📌 We break this into two mutually exclusive cases:

📌 Then, 2 more members are chosen from the remaining 6 members (since A and B are already selected):

📌 Then, all 4 members are chosen from the remaining 6 members:

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