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Home » Fandom.com Just Unlocked the HOT SECRET Wanting Fans Never Saw Coming! - Abbey Badges

Fandom.com Just Unlocked the HOT SECRET Wanting Fans Never Saw Coming! - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 \frac{63}{384} = \frac{21}{128}.

📌 Thus, the probability that the number of heads equals the die roll is $ \boxed{\dfrac{21}{128}} $.

📌 Question: An elementary school student is learning about the solar system and draws 5 planets, coloring each one either Earth-like (E) or Jupiter-like (J), but insists that no two adjacent planets in her linear diagram are both Jupiter-like. How many valid colorings are possible?

📌 Solution: We are to count the number of binary strings of length 5 (using 'E' and 'J', or 0 and 1), where no two adjacent characters are both 'J'. This is equivalent to counting binary strings without consecutive 1s.

📌 Let $ a_n $ be the number of valid colorings of $ n $ planets with no two adjacent Jupiter-like planets.

📌 This satisfies the Fibonacci recurrence:

📌 a_n = a_{n-1} + a_{n-2},

📌 with $ a_1 = 2 $ (E, J), $ a_2 = 3 $ (EE, EJ, JE) — JJ is invalid.

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P(2k) = P_0 \cdot 2^{2k/k} = P_0 \cdot 2^2 = 4P_0
Thus, the number of publications after \(2k\) years is:
\boxed{4P_0}
Question:** A deep sea microbiologist is studying the population growt...
P(t) = \frac{1000}{1 + 9e^{-0.5t}}
Find the time \(t\) when the population reaches half of its carrying c...
Solution:** The carrying capacity \(K\) of the logistic model is the m...
= \frac{1000}{1 + 9e^{-0.5t}}
Multiply both sides by the denominator:
500(1 + 9e^{-0.5t}) = 1000

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