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📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 5Question: A bioinformatician is analyzing sequences composed of nucleotides A, C, G, and T. How many distinct 8-nucleotide sequences contain exactly two A’s, three C’s, and three G’s (with no T’s), given that no two consecutive nucleotides are identical?

📌 Solution: We are to count the number of distinct 8-nucleotide sequences using exactly 2 A’s, 3 C’s, and 3 G’s (total 8 nucleotides), such that **no two identical nucleotides are adjacent**.

📌 This is a constrained permutation problem with repeated elements and adjacency restrictions.

📌 Let’s denote the multiset: {A×2, C×3, G×3}. Total length = 8. Total unrestricted arrangements (without adjacency restriction) would be the multinomial coefficient:

📌 \frac{8!}{2! \cdot 3! \cdot 3!} = \frac{40320}{2 \cdot 6 \cdot 6} = \frac{40320}{72} = 560

📌 But we must subtract or eliminate sequences where **identical nucleotides are adjacent**, particularly the two A’s and the three C’s or three G’s.

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