Find the number of paths in Z³ Find all paths in\begin{equation}  \mathbb Z^3 
\end{equation} of 6 steps where each step consists in increase or decrease by 1 one of the coordinates where it is and that start and finish in the (0,0,0) point.
I have been on this problem and my solution is based in dividing the problem in 3 cases: When it only moves in one axis, when it moves in 2 axis and when it moves in 3 axis. However I would be grateful if someone could give a shorter solution.
 A: Here is another looking at how many ways you can get to different positions after $3$ steps
$0$ steps
1

$1$ step
           1 
1      1   0   1      1
           1

$2$ steps
                              1
           2              2   0   2              2
1      2   0   2      1   0   6   0   1      2   0   2      1  
           2              2   0   2              2
                              1

$3$ steps
                                                         1  
                              3                      3   0   3
           3              6   0   6              3   0  15   0   3
1      3   0   3      3   0  15   0   3      1   0  15   0  15   0   1 ... 
           3              6   0   6              3   0  15   0   3
                              3                      3   0   3
                                                         1

and $6\times 1+ 24 \times 3+8 \times 6+6\times 15= 216 = 6^3$ as expected
but then you need to get back in the next three steps so the answer is
$$6\times 1^2+ 24 \times 3^2+8 \times 6^2+6\times 15^2= 1860$$
the same as JMoravitz's generating function method of looking at the constant in the expansion of $(x+x^{-1}+y+y^{-1}+z+z^{-1})^6$.
