Find the number of pathways from A to B if you can only travel to the right and down. I would like to solve the following, using Pascal's Triangle. Since there are shapes withing shapes, I am unsure as to where I should place the values. 
EDIT 1:

Where do I go from here? How do I get the value for the next vertex?
EDIT 2:

Okay, how does this look? From this, I would have 22 pathways. 
 A: This problem can be solved by labeling every vertex with the number of ways you can get to $B$ by traveling only right and down. Then the label at vertex $A$ is the desired number.
As to how you can do this labeling, work backwards from $B$. Starting at the vertex directly above $B$, there is only one way to get to $B$ (namely going down). So label this vertex with $1$. Similarly the vertex to the left of $B$ should have label $1$.
Now for the vertex two spaces above $B$, you can only move down. And if you move one space down, there is only one way to get to $B$; thus the vertex two spaces above $B$ should have label $1$.
For the vertex to the left and upwards of $B$, there are two possible paths: one to the right, and one down. If you take the path to the right, you end up at a label with one path to $B$, so there is one path going to the right. If you take the path down, there is a label with only one path, so only one path to down. So the label at this vertex is the sum of the labels to the right and downward of the vertex.
It's easy to reason that this last rule holds in general for all vertices. This should give you an inductive procedure to label backwards, all the way to $A$.
A: Since we can go from $A$ to $B$ in at most eigth steps, we can compute the number of paths by computing the eigth power of the adjacency matrix of our configuration with a self-loop in $B$.
A: You can do this algorithmically by labeling $B$ as $1$ and then repeating the following algorithm:

Choose some vertex such that every vertex reachable to the right or down from it is labeled. Label this by the sum of the label of the vertex to the right and the label of the vertex below (when these exist).

Which represents counting the number of paths from that vertex to $B$ since you can choose to either go down or right from any vertex. Eventually, you'll reach $A$ as you go backwards through the graph labeling things.
