<p><strong>Example 91:</strong> In how many ways the sum of upper faces of four distinct dice can be five?</p>
Step-by-Step Solution
Key Concept: Convert the problem of finding outcomes when rolling four dice with sum 5 into finding the number of integer solutions to a linear Diophantine equation, then use generating functions.
<p><strong>Step 1:</strong> The number of required ways equals the number of solutions of $x_1 + x_2 + x_3 + x_4 = 5$ where $1 \leq x_i \leq 6$ for $i = 1, 2, 3, 4$.</p><p><strong>Step 2:</strong> Since the upper limit is 6, which is greater than the required sum, we treat the upper limit as infinite.</p><p><strong>Step 3:</strong> The number of solutions equals the coefficient of $\alpha^5$ in the expansion of $(1 + \alpha + \alpha^2 + \cdots)^4$.</p><p><strong>Step 4:</strong> This equals the coefficient of $\alpha^5$ in $(1 - \alpha)^{-4}$.</p><p><strong>Step 5:</strong> Using the binomial series expansion: $^8C_5 = {^8C_3} = \frac{8 \cdot 7 \cdot 6}{1 \cdot 2 \cdot 3} = 56$</p><p>∴ Answer is <strong>56</strong>.</p>
Correct Answer: 56