Back to Robert's Math Figures

The number of possible paths of length 5

nfrom one corner of ann-by-n-by-n-by-n-by-nlattice to the opposite corner can be calculated using this formula:

The first few terms are 1, 120, 113400, 168168000, 305540235000, 623360743125120, ... (Compare this to the 2D, 3D, and 4D versions of the same idea.)

The empty 1-by-1-by-1-by-1-by-1 lattice looks like this, where each pair of "adjacent" points is joined by a line, and the starting and ending points are highlighted:

Each step of the path can occur in one of five directions.

For the 1-by-1-by-1-by-1-by-1 case, we're counting paths of length 5; since each path will be made up of one step in each of the five directions, we can easily enumerate the paths by computing all the permutations of (

dir_{1},dir_{2},dir_{3},dir_{4},dir_{5}).For the 1 x 1 x 1 x 1 x 1 lattice, here are the 120 (that is, 5!) paths:

Designed and rendered, at one time or another, using

Mathematicafor the Apple Macintosh, Microsoft Windows, and for NeXT.

Suggestion Box || Home || The Math Library || Help Desk || Quick Reference || Search