Find the number of possible solutions for $x,y,z$ for each the following cases.

$Case\ 1.$ Case of unlimited repetition.

$x + y +z = 10$ and $x \geq 0\ , y \geq 0,\ z \geq 0 $

$Case\ 2 $ Case of unlimited repetition with variable lower bounds

$x + y +z = 10$ and $x \geq 1\ , y \geq 2 \ , z \geq 3\ $

$Case\ 3 $ case when upper bounds and lower bounds are given , no variable can be greater than upper bound.

$x + y +z = 10$ and $10 \geq x \geq 0\ , \ 10 \geq y \geq 0 \ , 10 \geq z \geq 0\ $

$Case\ 4 $ case when upper bounds and lower bounds are given , lower bounds are different, no variable can be greater than upper bound

$x + y +z = 10$ and $10 \geq x \geq 1\ , \ 10 \geq y \geq 2 \ , 10 \geq z \geq 3\ $

$Case\ 5 $ case when upper bounds and lower bounds are given , only 1 variable can be greater than upper bound in the invalid case that gives correct sum like 11+2+3 = 16

$x + y +z = 16$ and $10 \geq x \geq 0\ , \ 10 \geq y \geq 0 \ , 10 \geq z \geq 0\ $

$Case\ 6 $ case when upper bounds and lower bounds are given , lower bounds are different ,only 1 variable can be greater than upper bound in the invalid case that gives correct sum like 11+2+3 = 16

$x + y +z = 16$ and $10 \geq x \geq 1\ , \ 10 \geq y \geq 2 \ , 10 \geq z \geq 3\ $

$Case\ 7 $ case when upper bounds and lower bounds are given , only 2 variable can be greater than upper bound in the invalid case that gives correct sum like 11+11+3 =25

$x + y +z = 25$ and $10 \geq x \geq 0\ , \ 10 \geq y \geq 0 \ , 10 \geq z \geq 0\ $

$Case\ 8 $ case when upper bounds and lower bounds are given , lower bounds are different ,only 2 variable can be greater than upper bound in the invalid case that gives correct sum like 11+11+3 =25

$x + y +z = 25$ and $10 \geq x \geq 1\ , \ 10 \geq y \geq 2 \ , 10 \geq z \geq 3\ $

$Case\ 9 $ case when upper bounds and lower bounds are given , lower bounds are different ,all variable can be greater than upper bound in the invalid case that gives correct sum like 11+11+11 = 33

$x + y +z = 33$ and $10 \geq x \geq 1\ , \ 10 \geq y \geq 2 \ , 10 \geq z \geq 3\ $

$Case\ 10 $ case when upper bounds and lower bounds are given , and both upper bound and lower bounds are variable.

$x + y +z = 10$ and $8 \geq x \geq 1\ , \ 20 \geq y \geq 2 \ , 12 \geq z \geq 3\ $