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Home » From \( P(2) = 960 \): - wispro

From \( P(2) = 960 \): - wispro

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 \boxed{\dfrac{5}{38}}

📌 Question: An ice cream shop offers 7 flavors and 4 toppings. How many ways are there to choose 3 distinct flavors and 2 distinct toppings?

📌 Solution: The number of ways to choose 3 flavors is $ \binom{7}{3} = 35 $, and 2 toppings is $ \binom{4}{2} = 6 $. Multiplying gives $ 35 \cdot 6 = 210 $ combinations.

📌 Question: How many ways are there to distribute 6 distinct marine samples into 3 identical containers, ensuring no container is empty?

📌 Solution: This is the Stirling number of the second kind $ S(6, 3) $, calculated as $ \frac{1}{3!} \left( 3^6 - 3 \cdot 2^6 + 3 \cdot 1^6 \right) = \frac{729 - 192 + 3}{6} = \frac{540}{6} = 90 $.

📌 \boxed{90}

📌 Question: A historical archive contains 4 ancient scrolls, 6 manuscripts, and 2 letters. If two items are drawn at random, what is the probability that both are manuscripts?

📌 Solution: Total items: $ 4 + 6 + 2 = 12 $. Total ways to choose 2: $ \binom{12}{2} = 66 $. Favorable ways: $ \binom{6}{2} = 15 $. Probability: $ \frac{15}{66} = \frac{5}{22} $.

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