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Level 2: 2 km/pixel - Coaching Toolbox

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Total unrestricted circular arrangements:

📌 Treat Alice and Bob as a single block. Since they must sit together and in a circle, we consider the block and the other 3 individuals—4 entities total. Number of circular arrangements of 4 entities: $ (4-1)! = 6 $.

📌 Within the block, Alice and Bob can switch places: 2 arrangements.

📌 Question: A digital archive system uses 6-digit access codes where each digit is from 0 to 9, but the first digit must be odd, the last digit must be even, and no digit repeats. How many valid access codes are possible?

📌 Solution: We are forming a 6-digit code with the following constraints:

📌 No further restrictions on middle digits

📌 We use a step-by-step digit assignment:

📌 However, we must ensure no reuse. Since $ d_1 $ is odd, $ d_6 $ (even) cannot conflict with it. So $ d_6 $: 5 choices

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