Number of paths in square grid (Project Euler problem 15)

Description of the problem (from here ):
Starting in the top left corner of a 22 grid, there are 6 routes (without backtracking) to the bottom right corner.

How many routes are there through a 2020 grid?


A: We can observe that:

•  every path has exactly 2*n moves, every move being either right or down (backtracking isn't allowed)
• in every 2*n path there are n moves down and n moves right. 

So, the number of all the distinct possibilities is the number of ways we can arrange n right moves from 2*n positions: