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Home » This Hellcat Logo Just Broke Contests — Here’s Why It’s Irresistible! - 500apps

This Hellcat Logo Just Broke Contests — Here’s Why It’s Irresistible! - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 Solution: Let the arrival times of Dr. Tide and Dr. Coral be $ x $ and $ y $, respectively, where $ x, y \in [0, 60] $ minutes after 7:00. The condition $ y > x $ defines a triangular region in the $ xy $-plane with area $ \frac{60^2}{2} = 1800 $. The favorable region is where $ x < 45 $ and $ y > x $. Integrating over $ x \in [0, 45] $, the area is $ \int_{0}^{45} (60 - x) \, dx = 60 \cdot 45 - \frac{45^2}{2} = 2700 - 1012.5 = 1687.5 $. The conditional probability is $ \frac{1687.5}{1800} = \frac{3}{4} $.

📌 \boxed{\dfrac{3}{4}}

📌 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?

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