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\[ = rac12 imes 13 \] - Ready Digital AB

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 Hence, the number of valid installation configurations is $ \boxed{10} $.

📌 Question: A deep-sea researcher collects 9 extremophile microorganisms near a hydrothermal vent, including 4 thermophiles, 3 psychrophiles, and 2 barophiles. If the researcher sequences DNA samples in a lab dish by selecting one organism per day for 9 consecutive days, with organisms of the same type indistinguishable, how many distinct sequencing orders are possible?

📌 Solution: We are arranging 9 organisms where there are repeated types: 4 thermophiles (T), 3 psychrophiles (P), and 2 barophiles (B). The number of distinct permutations of a multiset is given by:

📌 \frac{9!}{4! \cdot 3! \cdot 2!}

📌 9! = 362880,\quad 4! = 24,\quad 3! = 6,\quad 2! = 2

📌 \frac{362880}{24 \cdot 6 \cdot 2} = \frac{362880}{288} = 1260

📌 Thus, the number of distinct sequencing orders is $ \boxed{1260} $.

📌 Question: If a fair six-sided die is rolled 4 times, what is the probability that exactly two rolls show a prime number and the sum of all four rolls is even?

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