How many ways can I go from 1 to 10 in the following diagram?
You can think the problem as start with the $10$ and follow the numbers in order until you reach $1$.
Then there is two ways for each step: up or left. Then there are $2^{10-1}=2^9=512$ ways. Done!
Each square shows the number of (legal) paths to that square.
So the number of possible paths to the bottom right square is 512.