I don't quite understand the answer to this problem, and would like help explaining it, if my understanding isn't correct.
$$x_1 +x_2 +x_3 +x_4 < 40, x_1 > 0, x_2 > 0, x_3 \geq 0, x_4 > 5$$
I checked in the back of the book and it shows that it's $C(4+32-1,32)$ then says (the number of solutions to $x_1' +x_2' +x_3 +x_4' = 32$, where $x_1' = x_1 - 1$, $x_2'= x_2 - 1$, and $x_4'= x_4 - 6$)
This is what I think is going on, please correct my logic if I'm wrong.
$x_4' = x_4 - 6$ comes from the fact that $x_4 > 5$.
Then, $x_1' = x_1 - 1$ and $x_2' = x_2 - 1$ is because both $x_1$ and $x_2$ are $> 0$.
And since $x_3 \geq 0$, I don't have to worry about it since it can possibly be $0$.
Then it becomes
$$x_1 +x_2 +x_3 +x_4 +x_1 < 40-1-1-6 = 32$$