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Home » 2005 Finals Secrets Exposed: The Moment History Was Rewritten in Sports! - Databee Business Systems

2005 Finals Secrets Exposed: The Moment History Was Rewritten in Sports! - Databee Business Systems

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 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:

📌 $ b_n $: sequences ending in 3

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Solution:** Each step halves the risk:
After step 1: \( 64 / 2 = 32\% \)
After step 2: \( 32 / 2 = 16\% \)
After step 3: \( 16 / 2 = 8\% \)
After step 4: \( 8 / 2 = 4\% \)
Question:** A bioinformatician compares DNA sequences across 8 species...
Solution:** Number of ways to choose 2 out of 8:
\( \binom{8}{2} = \frac{8 \times 7}{2} = 28 \)
Solution:** Start: 1 atm at surface.
Each 100 m multiplies pressure by 1.1.

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