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Home » This uses the inclusion-exclusion principle: - 500apps

This uses the inclusion-exclusion principle: - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 \binom{7}{4} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

📌 So, the probability becomes:

📌 P(X = 4) = 35 \times \left( \frac{1}{2} \right)^7 = 35 \times \frac{1}{128} = \frac{35}{128}

📌 Thus, the probability that exactly 4 stations detect an earthquake is \(\boxed{\frac{35}{128}}\).

📌 Question:** A historian of science is cataloging 5 unique ancient manuscripts and 4 identical copies of a famous scientific treatise. In how many distinct ways can these 9 documents be arranged on a shelf?

📌 We have a total of 9 documents: 5 unique manuscripts and 4 identical copies of a treatise. The number of distinct arrangements of these items is given by the formula for permutations of multiset:

📌 \frac{n!}{k_1! \cdot k_2! \cdots k_m!}

📌 Here, \( n = 9 \) (total items), with 4 identical items (the copies), and the remaining 5 are distinct. So, the number of distinct arrangements is:

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