## Number of Possible Routes From Vertex to Opposite Vertex of n Dimensional Cube

Take a cube in
$\mathbb{R}^n$
with sides parallel to coordinate axes. How many paths are the from a corner to the opposite corner, such that coordinate axis is only travelled parallel to once? Starting from a corner we can move parallel to any of the
$n$
axes, and from the next corner to any of the
$n-1$
remaining axes and so on. The number of possible directions reduces by one with each successive vertex reached. Each axis must be travelled parallel to once, meaning the number of possible paths from vertex to opposite vertex is
$n!$
.