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Home » Factor out 2 in the numerator: - Sourci

Factor out 2 in the numerator: - Sourci

📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 Question: A meteorologist is analyzing storm patterns over a 7-day period, where each day is classified as either rainy, cloudy, or sunny. If exactly 3 days are rainy, 2 are cloudy, and 2 are sunny, how many different weather sequences are possible?

📌 Solution: We are assigning weather types to 7 days such that 3 are rainy (R), 2 are cloudy (C), and 2 are sunny (S). Since the days are distinct but the types are grouped, the number of distinct sequences is the multinomial coefficient:

📌 rac{7!}{3! \cdot 2! \cdot 2!} = rac{5040}{6 \cdot 2 \cdot 2} = rac{5040}{24} = 210

📌 Therefore, the number of possible weather sequences is $oxed{210}$.

📌 Question: A science communicator is creating a 6-part video series on fundamental particles, choosing episodes on quarks, leptons, bosons, photons, and neutrinos. If there are 2 quark episodes, 2 lepton episodes, and 2 boson episodes (each topic has identical content within its category), how many distinct orders can the videos be released in?

📌 Solution: We are arranging 6 video episodes where there are 2 identical quark episodes (Q), 2 identical lepton episodes (L), and 2 identical boson episodes (B). The number of distinct release sequences is given by:

📌 rac{6!}{2! \cdot 2! \cdot 2!} = rac{720}{2 \cdot 2 \cdot 2} = rac{720}{8} = 90

📌 Hence, the number of distinct release orders is $oxed{90}$.

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