Counting paths through a checkers board (only moving diagonally)

Question

a) Imagine that a checker is placed in the bottom right corner of a 6 by 6 checker board. The piece mmay be moved one square at a time diagonally left or right to the next row up. Calculate the number of different paths to the top row. b) repear part (a) with the checker placed in the bottom right corner.

Okay, so I tried using pascal's triangle for this. I calculted all the paths leading the the top row, then I added them (I got 1+5+10= 16) The textbook says the answer is 10.

I understand there is a way to do this with combinatorics, but I don't really understand it. Please explain to me in both ways, if possibble.

Answer 1

You can compute them from the number of left factors of Dyck paths, consisting of n steps.

For example for 6x6 board, there are 5 steps, the number of paths are given as

$$ {n \choose {\left \lfloor {n/2}\right \rfloor}}= {5 \choose 2} = 10 $$

Answer 2

One can count the number of ways to reach a square as follows:

The number of ways of reaching the top row is $1+4+5=10$.