Problem Solving Episode: Pascal's Method Part 1

In paying homage to my dedicated data management teacher who unfortunately got laid off in his first time teaching at my school, I present you the following problem:

A checker is placed on a checkerboard as shown. The checker may move diagonally upward. Although it cannot move into a square with a red square, the checker may jump over the X into the diagonally opposite square. How many path are there to the top of the board?

Understanding the Problem:
I am asked to determine the number of paths at the top of the board. This is calculated by determining the number of steps at each square.

Devise a Plan:

Pascal's method: Created by an iterative process, each term is equal to the term of the two terms immediately above it.


Carry out the plan:

Using Pascal's method, I will start by putting down the patterns that can only be made ONCE and considering the red squares as well which I cannot make a move:

Now, time to calculate the terms above by simply summing them: