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Home » S = 3 \cdot \frac(1/3)^21/3 = 3 \cdot \frac1/91/3 = 3 \cdot \frac13 = 1. - Veritas Home Health

S = 3 \cdot \frac(1/3)^21/3 = 3 \cdot \frac1/91/3 = 3 \cdot \frac13 = 1. - Veritas Home Health

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Alternatively, we recognize that this type of problem is best solved via **programming enumeration**, but since it's Olympiad, perhaps the answer is expected via **advanced combinatorial construction**.

📌 Wait — perhaps the intended method is **arranging the letters by generating valid permutations with constraints using factorials and multiplicative corrections**.

📌 But after extensive analysis, the correct and most efficient known method for such problems is **recursive backtracking with memoization**, but we must give a closed-form answer.

📌 Actually, let’s reframe using **the general formula for multiset permutations with no two adjacent identical elements** — but it’s complex.

📌 A known approach: use **the principle of first placing the most frequent elements with separation, then interleaving the rest**.

📌 Let’s try:

📌 We have to place 3 C’s and 3 G’s, no two identical adjacent. Also 2 A’s.

📌 First, place the 3 C’s with no two adjacent. As before, number of ways to choose 3 non-adjacent positions in 8 is $ \binom{8 - 3 + 1}{3} = \binom{6}{3} = 20 $.

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