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📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 This is a binomial probability with $ n = 5 $, $ k = 3 $, and success probability $ p = \frac{1}{3} $.

📌 P(X = 3) = \binom{5}{3} \left(\frac{1}{3}\right)^3 \left(\frac{2}{3}\right)^2 = 10 \cdot \frac{1}{27} \cdot \frac{4}{9} = 10 \cdot \frac{4}{243} = \frac{40}{243}

📌 Thus, the probability is $ \boxed{\frac{40}{243}} $.

📌 Question: In a biotech lab simulation, 4 identical red enzymes, 3 identical green enzymes, and 2 identical blue enzymes are deployed in a sequence. How many distinct sequences can be formed where all red enzymes are indistinct, green indistinct, and blue indistinct?

📌 Total number of enzymes: $ 4 + 3 + 2 = 9 $.

📌 We are arranging 9 objects where there are repeated types:

📌 \text{Number of distinct sequences} = \frac{9!}{4! \, 3! \, 2!}

📌 Therefore, the number of distinct sequences is $ \boxed{1260} $.Question: A technology consultant models data flow efficiency using the function $ f(\theta) = \sin^4 \theta - 4\sin^2 \theta \cos^2 \theta + \cos^4 \theta $. Find the minimum value of $ f(\theta) $ over all real $ \theta $.

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