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Home » \[ S_10 = 3(10)^2 - Silent Sales Machine

\[ S_10 = 3(10)^2 - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Number of ways to assign the roles:

📌 Choose which outcome occurs twice: 4 choices

📌 From remaining 3, choose 2 to occur once each: $\binom{3}{2} = 3$

📌 The last one occurs once? No — we have only four positions. If one occurs twice, two others once, that’s 2+1+1 = 4, so the fourth is covered. So the fourth outcome (the unused one) doesn’t need to be chosen — it’s determined.

📌 But when counting sequences, we need to assign positions.

📌 Standard multinomial: number of sequences with frequencies (2,1,1,0) is:

📌 \frac{4!}{2!1!1!0!} = 12

📌 For each such frequency pattern, number of ways to assign which word has which frequency.

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