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Home » Area of path = \( 144 - 100 = 44 \). - Treasure Valley Movers

Area of path = \( 144 - 100 = 44 \). - Treasure Valley Movers

📅 April 16, 2026 👤 scraface

Apr 16, 2026

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📌 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} $.

📌 \boxed{\dfrac{5}{22}}

📌 How many 8-digit positive integers have all digits either 3 or 4, and have exactly three consecutive 3's?

📌 To solve this, we can use a combinatorial approach. We want to form an 8-digit number using the digits 3 and 4, with exactly three consecutive 3's. Let's break it down:

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