There's a reaaaally easy way to do this without using even simpliest combinatorics
This is our path. We're allowed to move only right and bottom.
Now, I put a 1 on that point. What does it mean? It means there's only 1 way to go from that point to the point B.
I put another 1. There's only 1 way to go from that point to B too.
I summed them and wrote 2 on that point. This means there are only 2 ways to go from that point to B.
I placed all 1s..
And summed the other points..
OK, we completed the diagram Every number on the points indicates the number of the ways to go from that point, to the point B.
So, there are 70 ways to go from A to B.. Solved!
This diagram with the numbers are called Pascal's Triangle. Numbers on it are also a list of combinations.