Are you sure? Is this board not a counter-example if A starts on the right-most column and B starts on the second from the right:
x x x x x x x 8 9 x x x x x x 8 x 9 x x x x x 8 x x 9 x x x x 7 x x x 9 x x 7 x x x x x A B x x x x x x
edit: That being said, I made a mistake by following the claim from above that there are 10 possible paths, in fact there are only 9 cards in a row.
Are you sure? Is this board not a counter-example if A starts on the right-most column and B starts on the second from the right: