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Home » The particle changes direction where the derivative \( y' \) changes sign. - RTA

The particle changes direction where the derivative \( y' \) changes sign. - RTA

📅 May 9, 2026 👤 scraface

May 09, 2026

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📌 Question: A historian studies three scientific breakthroughs from a collection of 20, where 5 are pivotal. What is the probability that exactly two of the selected breakthroughs are pivotal?

📌 Solution: The total number of ways to choose 3 breakthroughs is $ \binom{20}{3} = 1140 $. The number of favorable outcomes is $ \binom{5}{2} \cdot \binom{15}{1} = 10 \cdot 15 = 150 $. The probability is $ \frac{150}{1140} = \frac{5}{38} $.

📌 \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}

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\[ z = \frac{-(1 + i) \pm (1 - i)}{2} \]
First solution:
\[ z = \frac{-1 - i + 1 - i}{2} = \frac{-2i}{2} = -i \]
Second solution:
\[ z = \frac{-1 - i -1 + i}{2} = \frac{-2}{2} = -1 \]
Thus, the solutions are \( z = -i \) and \( z = -1 \)
\(\boxed{z = -1 \text{ and } z = -i}\)
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