Number of integer solutions to the equation $x_1+x_2+x_3+x_4=100$ $x_1+x_2+x_3+x_4 = 100$ with
$1 \le x_1\le10$ 
$2\le x_2\le15$ 
$x_3\ge5$ 
$0\le x_4\le10$
Apparently this is the same as
$y_1 + y_2 + y_3 + y_4 = 92$ with
$y_1 \le 9$
$y_2 \le13$
$y_4 \le10$
I understand the $3$ conditions, but what has happened to $x_3$? 
I can see that it has been subtracted but what has happened to the conditions?

We are defining $y_1=x_1-1, y_2=x_2-2,y_3=x_3-5,y_4=x_4$ and demanding that all the $y$'s be $\ge 0$.  Once we defined $y_3=x_3-5$, there is no constraint on $y_3$ except that it be nonnegative, so it is not listed in the constraints.
