Passage: Four dice are rolled. Question: What is the probability...

Total number of outcomes when four dice are rolled is $6^4 = 1296$.

Let $x_1, x_2, x_3, x_4$ be the numbers appearing on the four dice. We need to find the number of solutions to the equation:

$x_1 + x_2 + x_3 + x_4 = 6$

where $1 \le x_i \le 6$ for $i = 1, 2, 3, 4$.

Let $x_i = y_i + 1$, where $y_i \ge 0$. Substituting this into the equation, we get:

$(y_1 + 1) + (y_2 + 1) + (y_3 + 1) + (y_4 + 1) = 6$

$y_1 + y_2 + y_3 + y_4 = 2$

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$.<br><br>Number of favorable outcomes = ^{2+4-1}C_{4-1} = ^{5}C_{3} = 10$.

The required probability is $\dfrac{10}{1296} = \dfrac{5}{648}$.

Answer: $\dfrac{5}{648}$