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📅 May 11, 2026 👤 scraface

May 11, 2026

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📌 $$Question: A graphic designer is creating a logo with a sequence of 9 icons, consisting of 4 stars, 3 moons, and 2 suns. How many distinct arrangements can be made if the icons of the same type are indistinguishable?

📌 Solution: We are asked to count the number of distinct permutations of a multiset. The total number of icons is $4 + 3 + 2 = 9$, with repetitions of each type. The number of distinct arrangements is given by the multinomial coefficient:

📌 Question: A primatologist observes a group of 8 primates—3 chimpanzees, 2 gorillas, and 3 orangutans—sitting in a line. If primates of the same species are indistinguishable, how many distinct seating arrangements are possible?

📌 Solution: We are arranging 8 individuals with repetitions: 3 chimpanzees, 2 gorillas, and 3 orangutans. The number of distinct arrangements is:

📌 Question: A historian of science is organizing a timeline display with 10 historical artifacts: 5 from the 17th century, 3 from the 18th century, and 2 from the 19th century. If artifacts from the same century are indistinguishable, how many unique chronological sequences can be formed?

📌 Solution: We are arranging 10 items with repetitions: 5 of one type, 3 of another, and 2 of a third. The number of distinct sequences is:

📌 Question: A graphic designer uses 7 color blocks in a pattern—2 red, 2 blue, 1 green, and 2 yellow—arranged in a row. If blocks of the same color are indistinguishable, how many unique patterns can be created?

📌 Solution: We compute the number of distinct permutations of a multiset:

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