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Home » Total time: \(3 + 4 = 7\) hours - 500apps

Total time: \(3 + 4 = 7\) hours - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 + 720 + 640 + 1008 + 1680 + 1344 = \boxed{5584}

📌 Final Answer:** $ \boxed{5584} $

📌 Question:** A fair 6-sided die is rolled 4 times. What is the probability that the sum of the rolls is exactly 14?

📌 We are rolling a 6-sided die 4 times and want the number of integer solutions to:

📌 x_1 + x_2 + x_3 + x_4 = 14, \quad \text{where } 1 \leq x_i \leq 6

📌 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:

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