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Home » Inside the Infernal Abyss: Hell Is Empty—and These Devils Will Devour You! - Abu Waleed Tea

Inside the Infernal Abyss: Hell Is Empty—and These Devils Will Devour You! - Abu Waleed Tea

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Thus, the number of valid fossil selections is $ \boxed{324} $.

📌 \boxed{324}

📌 Question: How many 6-digit positive integers have all digits either 3 or 7, and contain at least one pair of consecutive 3s?

📌 Solution: First, total number of 6-digit numbers with digits only 3 or 7: each digit has 2 choices, so $ 2^6 = 64 $. But 6-digit numbers cannot start with 0, but since digits are only 3 or 7, all 64 are valid.

📌 Now, subtract those that do **not** contain two consecutive 3s (i.e., no "33" in the number).

📌 Let $ a_n $ be the number of $ n $-digit sequences of 3 and 7 with no two consecutive 3s.

📌 This is a Fibonacci-type recurrence:

📌 Let $ a_n = b_n + c_n $, where:

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