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Home » You Won’t Believe What Happens When You End All J Words Forever! - Appcentric

You Won’t Believe What Happens When You End All J Words Forever! - Appcentric

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 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.

📌 However, due to the complexity of inclusion-exclusion over multiple repeated elements and spatial constraints, we instead model this using **inclusion with arrangement constraints** — specifically, placing the letters so that no two identical ones are adjacent.

📌 A better approach is constructive: build the sequence step-by-step, placing the letters while avoiding adjacent duplicates.

📌 But because we have repeated letters and exact counts, we use a **recursive or combinatorial placement strategy** with the principle of **arrangements with no adjacent duplicates** for multiset permutations.

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