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Home » Is Caoliu the Ultimate Game Changer? Watch This Revolutionary Moment - Appfinity Technologies

Is Caoliu the Ultimate Game Changer? Watch This Revolutionary Moment - Appfinity Technologies

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Let $ y_i = x_i - 1 $, so $ 0 \leq y_i \leq 5 $, and the equation becomes:

📌 y_1 + y_2 + y_3 + y_4 = 10

📌 We now count the number of non-negative integer solutions to this equation with each $ y_i \leq 5 $. First, count all non-negative integer solutions to $ y_1 + y_2 + y_3 + y_4 = 10 $ without restriction:

📌 \binom{10 + 4 - 1}{4 - 1} = \binom{13}{3} = 286

📌 Now subtract the number of solutions where at least one $ y_i > 5 $. Let’s use inclusion-exclusion.

📌 Let $ A_i $ be the set of solutions where $ y_i \geq 6 $. Set $ z_i = y_i - 6 $, then $ z_i + y_j + \cdots = 4 $, so:

📌 |A_i| = \binom{4 + 4 - 1}{4 - 1} = \binom{7}{3} = 35

📌 There are 4 such sets, so total for single violations: $ 4 \cdot 35 = 140 $

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