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the house that built me lyrics - Simpleprint

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Thus, there are $ \boxed{56} $ possible selections.

📌 Question: A Mars colony’s AI schedules two solar panel maintenance tasks at random times between 10:00 and 11:00. What is the probability that the second task starts at least 20 minutes after the first?

📌 Solution: Represent the times as $(x, y)$ in a 60-minute square. The condition $ y \geq x + 20 $ defines a triangular region with base $ 60 - 20 = 40 $ minutes. The area of this region is $ \frac{1}{2} \times 40^2 = 800 $. The total area is $ 60^2 = 3600 $. The probability is $ \frac{800}{3600} = \frac{2}{9} $.

📌 Thus, the probability is $ \boxed{\dfrac{2}{9}} $.

📌 Question: A piezoelectric road wear engineer needs to distribute 4 distinct sensors into 2 identical sections. How many ways can this be done if each section must contain at least one sensor?

📌 Solution: This is equivalent to partitioning 4 distinguishable objects into 2 non-empty indistinct subsets, calculated by the Stirling number $ S(4, 2) $. Using the recurrence:

📌 S(4, 2) = S(3, 1) + 2 \times S(3, 2) = 1 + 2 \times 3 = 7

📌 Thus, there are $ \boxed{7} $ valid distributions.Question: The average of $3u - 4$, $7u + 2$, and $4u - 1$ is ... If $u$ is a positive multiple of 3 and $u^2$ is less than 100, what is the average?

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الحل: لنأخذ أي أربعة أعداد صحيحة متتالية: $n, n+1, n+2, n+3$.
يوجد على الأقل عددان زوجيان، وأحدهما قابل للقسمة على $4$، لذا يكون حاص...
يوجد على الأقل عدد قابل للقسمة على $3$.
لذلك، يكون حاصل الضرب قابلاً للقسمة على $8 \cdot 3 = 24$.
نختبر مثالاً: $1 \cdot 2 \cdot 3 \cdot 4 = 24$، والذي ليس قابلاً للقسم...
إذن، أكبر عدد صحيح يجب أن يقسم حاصل ضرب أي أربعة أعداد صحيحة متتالية ه...
السؤال: ما هي أكبر قيمة ممكنة للقاسم المشترك الأكبر $(a,b)$، إذا كان م...
الحل: ليكن $d = \gcd(a, b)$. إذن $a = dx$، $b = dy$، حيث $\gcd(x, y) =...
ثم $a + b = d(x + y) = 96$، لذا يجب أن يكون $d$ قاسمًا للعدد $96$.
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