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2 1 2 gallon water jug - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 = 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: \(3! = 6\),

📌 So total: \(25 \times 6 = 150\).

📌 Thus, the number of distinct schedules is \(\boxed{150}\).Question: How many of the 100 smallest positive integers leave a remainder of 2 when divided by 5?

📌 Solution: We are looking for integers $ n $ such that $ n \equiv 2 \pmod{5} $ among the first 100 positive integers.

📌 The sequence of such numbers is $ 2, 7, 12, \dots, $ forming an arithmetic progression with first term 2 and common difference 5.

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