Passage: Four dice are rolled. Question: What is the probability that getting a total of the numbers on the dice is 6 ?

Step-by-step solution

Total number of outcomes when four dice are rolled is 6^4 = 1296 . Let x₁, x₂, x₃, x₄ be the numbers appearing on the four dice. We need to find the number of solutions to the equation: x₁ + x₂ + x₃ + x₄ = 6 where 1 x_i 6 for i = 1, 2, 3, 4 . Let x_i = y_i + 1 , where y_i 0 . Substituting this into the equation, we get: (y₁ + 1) + (y₂ + 1) + (y₃ + 1) + (y₄ + 1) = 6 y₁ + y₂ + y₃ + y₄ = 2 The number of non-negative integer solutions to this equation is given by ^ n+r-1 C_ r-1 , where n=2 and r=4 . Number of favorable