The 4x4 grid with unit cell and point A(0,0) is given. One can move up and to the right only. Find the number of paths of length 6 on this grid

Question

(a path of the length of 6 is the path with 6 steps on the grid starting at point A)

Answer 1

This is a problem in combinatorics.

There are 6 boxes (steps).

The boxes are to be filled with the letters R (right) and U (up).

There are only 3 ways of filling the boxes: 2R + 4U, 3R + 3U, and 4U and 2R.

There are ₆C₂ ways of putting 2 R's in the 6 boxes (and the same number of ways of putting 4 U's in 6 boxes);

15 ways with 2 R's and 15 ways with 2 U's.

With 3 and 3 there are ₆C₃ ways...20 ways.

Total paths: 15+20+15=50