How many integer solutions to the equation (discrete math)

Question

How many integer solutions to the equation $x_1 + x_2 + x_3 = 15$ so that $x_1 \ge 3, x_2 \ge 2$ and $x_3 \ge 0$ ? I honestly don't know where to start on this

Thanks

Answer 1

HINT: Let $y_1=x_1-3$, $y_2=x_2-2$, and $y_3=x_3$. Note that $x_1+x_2+x_3=15$ if and only if $y_1+y_2+y_3=10$, and integers $x_1,x_2$, and $x_3$ satisfy the inequalities $x_1\ge 3$, $x_2\ge 2$, and $x_3\ge 0$ if and only if the corresponding $y_1,y_2$ and $y_3$ are non-negative. Now use the standard stars-and-bars solution to the problem of counting solutions to $y_1+y_2+y_3=10$ in non-negative integers.