Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is $X/1296$. The value of $X$ is _____.

Let the outcomes of the four dice be $d_1, d_2, d_3, d_4$, where $1 \le d_i \le 6$ for each $i \in \{1, 2, 3, 4\}$. We are looking for the number of integer solutions to:

By symmetry, the number of ways to get a sum of $S$ when rolling 4 six-sided dice is equal to the number of ways to get a sum of $4(1 + 6) - S = 28 - S$. For $S = 22$:

Alternatively, let $y_i = 6 - d_i$, where $0 \le y_i \le 5$. Then:

Since each $y_i \ge 0$ and the sum is $2$, the condition $y_i \le 5$ is automatically satisfied for all $i$.

The number of non-negative integer solutions to $y_1 + y_2 + y_3 + y_4 = 2$ is given by the stars and bars formula:

Thus, the number of favorable outcomes is $X = 10$.

The probability is $\frac{10}{6^4} = \frac{10}{1296}$, so $X = 10$.