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How many integer solutions are there to the system of inequalities
$$x_{1}+x_{2}+x_{3}+x_{4} \leq 15, \quad x_{1}, \ldots, x_{4} \geq 0 ?$$

We add one extra variable to turn the inequality to an equality:
$$x_{1}+x_{2}+x_{3}+x_{4}+s=15, \quad s, x_{i} \geq 0 .$$
This gives $\left(\begin{array}{c}15+5-1 \\ 5-1\end{array}\right)$ integer solutions.