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\( n = 1 \) works. - IX Labs

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question: A science communicator creates a 10-part video series with 4 documentaries, 3 animations, and 3 interviews. If the order of video types matters but identical types are indistinct, how many unique sequences can be arranged?

📌 Solution: This is a multinomial permutation problem. The total sequences are given by:

📌 \frac{10!}{4! \cdot 3! \cdot 3!} = \frac{3628800}{24 \cdot 6 \cdot 6} = \frac{3628800}{864} = 4200

📌 The number of unique video sequences is $\boxed{4200}$.

📌 Question: Three insect species are randomly selected from 10 bee species, 6 butterfly species, and 4 moth species. What is the probability that exactly two are bees and one is a butterfly?

📌 Solution: First, compute the total number of ways to choose 3 species: $\binom{20}{3} = 1140$. The number of favorable outcomes is $\binom{10}{2} \cdot \binom{6}{1} = 45 \cdot 6 = 270$. The probability is:

📌 \frac{270}{1140} = \frac{9}{38}

📌 The probability is $\boxed{\dfrac{9}{38}}$.

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