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📅 May 8, 2026 👤 scraface

May 08, 2026

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📌 ### Step 2: Count the number of favorable outcomes

📌 We need to count the number of 6-digit sequences where exactly one pair of consecutive digits is equal, and all other adjacent digits are different.

📌 There are 5 possible positions for a pair of consecutive digits: positions (1,2), (2,3), (3,4), (4,5), or (5,6).

📌 For each such position $ i $, we choose the digit at $ d_i $ and $ d_{i+1} $ to be the same (3 choices), and ensure that $ d_{i-1}

📌 eq d_i $ and $ d_{i+2}

📌 eq d_{i+1} $, provided those indices exist.

📌 Letâs fix a position $ i $, and count the number of sequences where the repeated pair is at $ i, i+1 $, and no other consecutive digits are equal.

📌 We proceed as follows for one fixed $ i $:

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