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Home » Anything But a Backpack Day? Impossible—Here’s Why You’ll Fail Hands-Free! - AssociationVoting

Anything But a Backpack Day? Impossible—Here’s Why You’ll Fail Hands-Free! - AssociationVoting

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 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:

📌 \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.

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