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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: A climate scientist is modeling sea level rise scenarios using binary strings of length 8, where each bit represents a coastal zone either flooded (1) or not flooded (0). How many such strings have at least three consecutive 1s?

📌 Total number of binary strings of length 8:

📌 Let $a_n$ be the number of binary strings of length $n$ with no three consecutive 1s.

📌 If the string ends in 0, the previous $n-1$ bits can be any valid string of length $n-1$.

📌 If it ends in 10, the previous $n-2$ bits can be any valid string of length $n-2$.

📌 If it ends in 110, the previous $n-3$ bits can be any valid string of length $n-3$.

📌 Number of strings with at least three consecutive 1s:

📌 Question: A tech consultant is analyzing password strength by counting how many 6-character passwords (using lowercase letters and digits) contain at least one repeated character.

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