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what day does easter fall on - DevRocket

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Solution: This is a multinomial coefficient problem. We are arranging 8 actions where 3 are of one kind, 3 of another, and 2 of a third, all indistinct within types. The number of distinct sequences is:

📌 Question: A UI designer is assigning 6 different color themes to 3 identical layout palettes, with each palette receiving at least one theme. How many ways can the themes be distributed across the palettes, considering palettes are indistinguishable?

📌 Solution: We seek the number of ways to partition 6 distinguishable themes into 3 non-empty, unlabeled subsets. This is given by the Stirling number of the second kind $S(6, 3)$. From known values:

📌 Thus, there are 90 distinct distributions of themes across the identical palettes.

📌 Question: An entomologist observes that in a sample of 10 insects, 4 are butterflies, 3 are bees, and 3 are beetles. If a research team randomly selects 4 insects to tag, what is the probability that exactly 2 are bees and at least one is a butterfly?

📌 Solution: First, compute the total number of ways to choose 4 insects from 10:

📌 Now, count favorable outcomes: exactly 2 bees and at least 1 butterfly. Since 2 bees are fixed, the remaining 2 insects come from 4 butterflies and 3 beetles (7 insects), but excluding cases with no butterflies.

📌 So, valid combinations: $21 - 3 = 18$

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