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How many integer solutions does the equation
$$x_{1}+x_{2}+x_{3}+x_{4}=15$$
have, if we require that $x_{1} \geq 2, x_{2} \geq 3, x_{3} \geq 10$ and $x_{4} \geq-3 ?$

Let $y_{1}=x_{1}-2, y_{2}=x_{2}-3, y_{3}=x_{3}-10$ and $y_{4}=x_{4}+3$.
We get a new equation where $y_{i} \geq 0$ and
\begin{aligned} \left(y_{1}+2\right)+\left(y_{2}+3\right)+\left(y_{3}+10\right)+\left(y_{4}-3\right) &=15, \quad \Leftrightarrow \\ y_{1}+y_{2}+y_{3}+y_{4} &=3 \end{aligned}
Dots and bars gives $\left(\begin{array}{c}3+4-1 \\ 4-1\end{array}\right)$ integer solutions.