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Home » At the End of the Earth: Cape Expedition 2022s Race Against Ice and Isolation - Anadea

At the End of the Earth: Cape Expedition 2022s Race Against Ice and Isolation - Anadea

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 Question: A linguist is analyzing a 6-letter word composed only of the letters A and B, where no two As are adjacent. How many such words are possible?

📌 We want to count the number of 6-letter words using only A and B, such that no two As are adjacent.

📌 Lets denote the number of As in the word as $k$. Since no two As can be adjacent, we must place the $k$ As in such a way that there is at least one B between them.

📌 To place $k$ As with no two adjacent, we consider placing $k$ As in a way that they occupy positions with gaps between them. This is equivalent to placing $k$ As in a sequence of $6 - k$ Bs such that As are separated.

📌 This is a classic stars and bars problem with restrictions. The number of valid placements of $k$ As in 6 positions with no two adjacent is equal to the number of ways to choose $k$ positions among the 6 such that no two are consecutive. This is given by:

📌 Question: A biologist is analyzing a DNA sequence of length 8, where each position is either A, T, C, or G. How many such sequences contain exactly two As and no two As are adjacent?

📌 We want to count the number of 8-letter sequences using letters A, T, C, G, such that exactly two As appear and no two As are adjacent.

📌 Step 1: Choose positions for the two As such that they are not adjacent. In 8 positions, the number of ways to choose 2 non-adjacent positions is:

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