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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: \(3! = 6\),
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.