Finding Nonnegative Solutions Using Generating Functions

Find the number of non-negative solutions to $x_1 + x_2 + x_3 = 6$, where $x_1 \leq 4$, $x_2 \leq 3$, and $x_3 \leq 5$ using generating functions.

In this problem, we are tasked with finding the number of non-negative integer solutions to a linear equation with constraints on individual terms using the method of generating functions. Generating functions are a powerful tool in combinatorics that allow us to encapsulate an entire sequence of numbers (such as the number of ways to achieve certain sums) into a single function.

The key challenge here is to respect the given constraints ($x_1 \leq 4$, $x_2 \leq 3$, $x_3 \leq 5$) while finding solutions. To tackle this, we reinterpret each constraint as a modification in the generating function. Essentially, for each variable $x_i$ with constraint max value $m_i$, we create a 'local' generating function that translates this into a polynomial of degree $m_i+1$ with equal coefficients, like $(1+x+x^2+...+x^{m_i})$. This setup effectively ensures no solution will exceed an individual variable constraint.

After setting the scene with generating functions for each variable, the problem of finding the number of solutions becomes one of calculating the coefficient of $x^6$ in the product of these local generating functions. This approach integrates key concepts of combinatorial analysis and algebraic manipulation, providing a deep understanding of both the constraints and the structural behavior of solutions. Additionally, this problem exemplifies the broader application of generating functions in solving constrained counting problems, which is a vital area of study in discrete mathematics.