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Home » \[ 37p + 7q + r = 45 \quad \text(E3) \] - Get link 4share

\[ 37p + 7q + r = 45 \quad \text(E3) \] - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 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?

📌 Solution: First, identify the prime numbers on a six-sided die: 2, 3, 5 → 3 primes; non-primes: 1, 4, 6 → 3 non-primes.

📌 Exactly two of the four rolls to be prime

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