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Question: What is the base-ten number represented by the base-nine number $ 243_9 $?
Solution: The base-nine number $ 243_9 $ means:
Solution: A lattice point has integer coordinates $ (x, y) $. The ellipse has semi-axes $ a = 5 $, $ b = 4 $.
We check integer pairs $ (x, y) $ satisfying this.
Solution: First, we calculate the total number of ways to select and arrange 5 manuscripts from 10, ensuring that at least 2 are astronomy-related. We consider three valid cases: exactly 2 astronomy manuscripts, exactly 3, and exactly 4.
Case 1: 2 astronomy, 3 non-astronomy
Arrange the 5 distinct manuscripts: $5! = 120$
Case 2: 3 astronomy, 2 non-astronomy
Case 3: 4 astronomy, 1 non-astronomy
Question: A science journalist is analyzing data from 12 experiments—7 on quantum mechanics and 5 on classical physics. She wishes to feature a sequence of 6 experiments in her article such that no two quantum experiments are adjacent. How many such sequences are possible if all experiments are distinct?
