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GO Classes Weekly Quiz 13 | Discrete Mathematics | Combinatorics | Question: 6
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How many integer solutions are there to the system of inequalities
$$
x_{1}+x_{2}+x_{3}+x_{4} \leq 15, \quad x_{1}, \ldots, x_{4} \geq 0 ?
$$
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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.
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