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Home » Substitute $ a = 4 $ into the equation: - inBeat

Substitute $ a = 4 $ into the equation: - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 4x - 2x + 1 + 3x = (4x - 2x + 3x) + 1 = 5x + 1.

📌 Solution: We are to count the number of 5-character strings using the letters A, B, C, D (4 letters), such that each letter appears at least once. This is a classic inclusion-exclusion problem.

📌 We first compute the total number of strings of length 5 using 4 letters:

📌 Now subtract the strings that miss at least one letter. Let $ S $ be the set of all such strings, and for each letter, let $ A_i $ be the set of strings that do not contain letter $ i $. We want:

📌 Missing all four letters: impossible, so 0.

📌 Alternatively, since there are 4 letters and we require each to appear at least once in 5 positions, exactly one letter appears twice, and the others once.

📌 Assign the remaining 3 letters to the 3 positions (in $ 3! = 6 $ ways)

📌 Count how many integers fall in each residue class:

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