Question

A person, standing initially at point p, starts walking. At each step, the person moves exactly 1 foot in one of the four possible directions: north, south, east or west, with equal probability. What is the probability that after 8 steps, the person returns to the point p?

Answer 1

if he takes one step in each four directions and takes one step back from each four directions then there is the probability that after 8 steps the person returns to the point