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do you lose an hour of sleep in the spring - Decision Point

📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 + 2025 + 2027 + 2029 = (2023 + 2029) + (2025 + 2027) = 4052 + 4052 = 8104

📌 Now compute $ 8104 \mod 8 $. Since 8100 is divisible by 8 (as $ 8100 \div 8 = 1012.5 $, but $ 8000 \div 8 = 1000 $, and $ 100 \div 8 = 12.5 $), we check only the last three digits:

📌 \div 8 = 13 \text{ exactly}, so } 8104 \equiv 0 \pmod{8}

📌 $$Question: How many ways are there to distribute 4 distinct chemical samples into 2 identical storage containers such that each container has at least one sample?

📌 Solution: The problem requires counting the number of ways to partition 4 distinct items into 2 non-empty identical subsets. This is given by the Stirling numbers of the second kind, $ S(4, 2) $. The formula for $ S(n, k) $ is $ S(n, k) = S(n-1, k-1) + k \cdot S(n-1, k) $. Using known values, $ S(4, 2) = 7 $. Thus, the number of ways is $ \boxed{7} $.

📌 Question: A philosopher examines 5 ethical guidelines. What is the probability that exactly 2 out of 3 randomly selected guidelines prioritize human welfare, if 3 guidelines are welfare-focused and 2 are not?

📌 Solution: The total number of ways to choose 3 guidelines from 5 is $ \binom{5}{3} = 10 $. The number of favorable outcomes (2 welfare, 1 non-welfare) is $ \binom{3}{2} \cdot \binom{2}{1} = 3 \cdot 2 = 6 $. The probability is $ \frac{6}{10} = \frac{3}{5} $. Thus, the probability is $ \boxed{\dfrac{3}{5}} $.

📌 Question: How many distinct 6-letter arrangements can be formed from the letters of "THEORETICAL" if the letters 'E' and 'A' must not be adjacent?

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