Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner?

Starting in the top left corner of a 2 x 2 grid, there are 6 routes (without backtracking) to the bottom right corner.

How many routes are there through a 20 x 20 grid?

**********

After doing a bit of research on this problem, it seems as though a simple Combinations formula will provide the answer. We know that any path will have 40 moves (20 right + 20 down), so for C(n,r) – n will equal 40 and r (the number of right moves) will equal 20 – C(40,20). Then we just plug the numbers into the formula – the ! means ‘factorial’ which is imported from the math class in python.

n!

C(n,r) = ———-

r!*(n-r)!

The number of how many good routes we have can be found by finding how

many combinations of 20 R’s we can have in our 40 moves, so we want to

calculate:

40!

C(40,20) = ———-

20!*(40-20)!

Helpful Links:

### One Possible Solution: Python

# Python version = 2.7.2 # Platform = win32 # Elapsed Time: 20.9999084473 millisecs from math import factorial import time def main(): """Main Program""" start_time = time.time() n = 40 # The total number of moves for any one path (right + down) r = 20 # The total number of right moves for any one path print factorial(n) / (factorial(r) * factorial(n - r)) print "Elapsed Time:", (time.time() - start_time) * 1000, "millisecs" if __name__ == '__main__': main()