How to solve this combinatorics problem? 
As a tourist in NY, I want to go from the Grand Central Station (42nd street and 4th Avenue) to Times Square (47th street and 7th Avenue). I needed my morning coffee, and wanted to go to a Starbucks that's located at 44th street and 5th avenue.
If I only walk West and North, how many ways are there for me to get there?

To get to the coffee you have $3$ choices, just decide when you going west.
Now from your coffee, you have $4$ ways to take the first west, $a_0,a_1,a_2,a_3$. Here $a_j$ means that your first move west is in the $j+1$ move.
If you take $a_0$ you are left with $4$ possibilities, if you take $a_1$ you have $3$...
So you have $4+3+2+1=10$ to get from the coffee to times square.
And the total number of ways is $3\cdot 10=30$.
