Number of integral solutions for $|x | + | y | + | z | = 10$ How can I  find the number of integral solution to the
equation
$|x | + | y | + | z | = 10.$
I am using the formula,
Number of integral solutions for $|x| +|y| +|z| = p$ is $(4P^2) +2 $, So the answer is 402.
But, I want to know, How we can find it without using formula.
any suggestion !!!
Please help

(1) z=0, $40$ patterns
z=1, $36$ 
z=2, $32$
・・・
z=9,$4$
but z=10,$2$ the total is
$S=2(4+8+\dots+36)+40+2=402$
(2) Another counting
There are 8 areas by plus and minus of x,y,z, 40 patterns at x=0, 2 patterns at y=z=0, therefore  
$40+8*10C2+2=402$
