How can I find the number of whole number solutions for $ x_1 + x_2 + x_3 + x_4 = 11 $? Find the number of whole number solutions for this equation
$ x_1 + x_2 + x_3 + x_4 = 11 $
where
$ x_1 \ge -2, x_2 \ge -1, x_3 \ge 0, x_4 \ge 5  $

*$ x_1 + x_2 + x_3 + ...+x_k = n \to \left(\begin{array}{c}n+k-1\\ k-1\end{array}\right) $
where
$ x_1 \ge -2, x_2 \ge -1, x_3 \ge 0, x_4 \ge 5  $
$$ x_1 \ge -2 \to  y_1=x_1 +2\ge 0\\
x_2 \ge -1 \to y_2=x_2 +1\ge 0\\
x_3 \ge 0\\
x_4 \ge 5 \to y_4=x_4 -5\ge 0$$
$$\color{red} {x_1+x_2+x_3+x_4=11\\(x_1+2)+(x_2+1)+(x_3)+(x_4-5)=11+2+1-5=9\\y_1+y_2+x_3+y_4=9} \to \left(\begin{array}{c}9+4-1\\ 4-1\end{array}\right)=\\\left(\begin{array}{c}12\\ 3\end{array}\right)=\frac{12.11.10}{3.2.1}=220$$
