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📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 5Question: An astronomer studies a cluster of 12 rogue planets initiated by gravitational interactions, 5 of which exhibit unusual radio emissions. If 4 planets are randomly selected for detailed spectral analysis, what is the probability that exactly 2 of them emit radio signals?

📌 Question: A primatologist observes a group of 9 orangutans, 3 of whom are dominant males. If a subgroup of 5 is chosen at random to study vocal communication patterns, what is the probability that at least one dominant male is included?

📌 Solution: We use complementary counting. First, compute the total number of ways to choose 5 orangutans from 9:

📌 So, the number of favorable outcomes (at least one dominant male) is:

📌 Question: A linguist analyzes a symbolic language with a 7-letter word composed of the letters: A, A, B, C, C, C, D—where A, C, and D appear multiple times, but B only once. How many distinct permutations of this word are possible?

📌 Solution: The total number of distinct permutations of a multiset is given by dividing the factorial of the total number of letters by the product of the factorials of the counts of each repeated letter. Here, there are 7 letters in total:

📌 So the number of distinct permutations is:

📌 Question: A primatologist records vocalizations from 6 distinct orangutans, assigning each a unique identification code of the form ABCDEF, but permutes only three specific digits—A, B, and C—while the rest remain fixed. In how many ways can the system generate a valid code if A must come before B?

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