5: Even Direction Changes Alice is navigating a 88 grid. She starts at the bottom-left corner and wants to reach the top-right corner. At each step, she moves either right (R) or up (U). Every valid path consists of exactly 8 R moves and 8 U moves arranged in some order. A direction change occurs whenever Alice switches from R to U or from U to R. Assume all valid paths are equally likely. What is the probability that a randomly chosen valid path has an even number of direction changes? Hint: Note that a path has an even number of direction changes if and only if it starts and ends with the same move