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5Question: A historian is examining 5 ancient star charts, 2 of which are identical copies of the same constellation map. How many distinct ways can they arrange these charts on a shelf?
Solution: The number of distinct arrangements is calculated by dividing the factorial of the total number of items by the factorial of the number of identical items. Here, there are $5!$ total permutations, but since 2 charts are identical, we divide by $2!$. Thus, the number of distinct arrangements is $
Question: A materials scientist is designing a high-temperature superconductor with 4 layers, each requiring one of 3 specialized alloys. If the order of layers matters and no two adjacent layers can use the same alloy, how many valid configurations are possible?

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