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5Question: How many distinct sequences can be formed by arranging 5A, 3B, and 2C flags if no two C flags are adjacent?
Solution: First, arrange the non-C flags (5A and 3B) which total $5 + 3 = 8$ flags. The number of distinct arrangements is $
Question: An archaeologist discovers 6 distinct pottery shards, 4 statues, and 3 gemstones. How many ways can they arrange these items in a row if all gemstones must be placed together?
Solution: Treat the 3 gemstones as a single super object. This gives $6 + 4 + 1 = 11$ items to arrange, with the 6 pottery shards and 4 statues being distinct. The number of arrangements is $
Question: A climatologist records 5 days of temperature data, each day classified as cold, moderate, or hot. How many sequences have exactly 2 cold days and no two hot days consecutive?
Question: A micropaleontologist measures stable isotope ratios of $1.2$, $1.8$, and $2.4$ in microfossils. What is the average ratio?
