Number of solutions of $|x|+|y|+|z|=10$ where $x,y,z\in\mathbb{Z}$ I am trying to find the number of solutions to the integral equation $|x|+|y|+|z|=10$ where $x,y,z\in\mathbb{Z}$. Here's what I've done so far.

My Attempt:
Let us fix a value of $|x|=k$, so $|y|+|z|=n-k$, (considering the general case of when $|x|+|y|+|z|=n$). So for each case $|y|+|z|=n-k$ has $(n-k+1)$ solutions in $|y|$ and $|z|$, and in total $4(n-k+1)$ because there's $4$ arrangements of $y$ and $z$ being $-$ or $+$, which is itself in correspondence with $2$ possibilities of $x$ when $|x|=k$, so total number of solutions should be $8\sum_{k=0}^{n}(n-k+1)=8\sum_{k=0}^{n}(k+1)=4(n+1)(n+2)$, so in this case number of solutions should be $528=48\cdot 11$.

The actual answer is given to be $402$. Any hints are appreciated. Thanks.  
 A: It suffices to count the non-negative solutions, i.e. $x,y,z\geq 0$ because if $x$ (or $y$ or $z$) is a solution, so is $-x$. Consider $3$ cases.
Case $1$: $x,y,z$ are positive. Using the stars and bars method, the number of triples is ${9\choose 2}=36$.
Case $2$: One of $x,y,z$ is $0$, and the other two are positive. Take $x=0$, for instance, to see that $y+z=10$ has $9$ positive solutions. So we have $9\cdot 3=27$ triples in this case.
Case $3$: One of $x,y,z$ is $10$, and the others are $0$. There are $3$ such triples.
In each triple from case $1$, any of the numbers can be taken with a positive or negative sign. In the triples from case $2$, this applies to exactly $2$ of the numbers (since the third one is $0$), and in the triples from case $3$, this applies only to the non-zero number. Hence, the total count is
$$2^3\cdot 36+2^2\cdot 27+2^1\cdot 3=402 $$ 
A: Hint:
Note that $+0 = -0 = 0$, i.e, $0$ doesn't have separate positive & negative values. When you're counting the number of solutions for the case $|y| + |z| = n - k$, you need to account for this (i.e., don't double count the $0$ values as being positive or negative) when you're determining the total number of solutions. I'll leave it to you to account for this to determine the appropriate updated total value.
