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Home » Common difference $ d = 2 $ - DNSFLEX

Common difference $ d = 2 $ - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 The simplified value is $ oxed{7} $.

📌 Question: A linguist uses a polynomial $ P(n) = n^3 + 3n^2 - 2n + 1 $ to model language complexity. Find the remainder when $ P(n) $ is divided by $ n - 1 $.

📌 Solution: By the Remainder Theorem, substitute $ n = 1 $ into $ P(n) $:

📌 $$ P(1) = 1^3 + 3(1)^2 - 2(1) + 1 = 1 + 3 - 2 + 1 = 3. $$

📌 The remainder is $ oxed{3} $.Question: A linguist is studying the evolution of 5 distinct dialects and wishes to group them into 3 indistinguishable historical clusters, where each cluster must contain at least one dialect. How many ways can this be done?

📌 We are asked to count the number of ways to partition 5 distinguishable items (dialects) into 3 non-empty, indistinguishable subsets (clusters). This is a classic problem in combinatorics involving **Stirling numbers of the second kind** and accounting for indistinguishable groups.

📌 Letâs denote the number of ways to partition $ n $ distinguishable objects into $ k $ non-empty, indistinguishable subsets as $ S(n, k) $, the **Stirling number of the second kind**.

📌 Here, $ n = 5 $, $ k = 3 $, so we compute $ S(5, 3) $.

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