Chess Board counting problem

Question

Consider a n x n chess board. Count the number of shortest paths from the position (0,0) to the position (100,100) if each move can either be a horizontal step or a vertical step. You may assume that n is larger than 100.

Answer 1

Each move can be either $(x,y)\to(x+1,y)$ (we call it "h") or $(x,y)\to(x,y+1)$ (we call it "v") so a shortest path is sequence of $100$ "h" and $100$ "v" mixed up in $$\frac{200!}{100!\cdot 100!}={200\choose 100} \hbox{ ways.}$$