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Home » \( a(a + b) = 20 \) - jntua results

\( a(a + b) = 20 \) - jntua results

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Question: A linguist analyzing word evolution models a simplified language where each word is a sequence of 8 characters, each chosen independently from the set $\{a, b, c\}$. What is the probability that a randomly generated word contains exactly 3 $a$âs, 2 $b$âs, and 3 $c$âs?

📌 Solution: This is a multinomial probability. Let $ X = (X_a, X_b, X_c) $ be the counts of each letter in a random 8-character word, with each character independently chosen from $\{a, b, c\}$ with equal probability $ rac{1}{3} $.

📌 P(X_a = 3, X_b = 2, X_c = 3) = rac{8!}{3! \cdot 2! \cdot 3!} \cdot \left(rac{1}{3}

📌 ight)^3 \cdot \left(rac{1}{3}

📌 ight)^2 \cdot \left(rac{1}{3}

📌 ight)^3 = rac{8!}{3! \cdot 2! \cdot 3!} \cdot \left(rac{1}{3}

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

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