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?
 A: I agree with Ofir's answer and would like to suggest an alternative, more general way to compute the number of ways to go from the cafe to Times square.
Clearly, the total number of steps is $5$. Clearly, exactly two of those steps are in the direction west. You just need to choose these two steps, and the number of possible choices is $$\left(\begin{array}{c}5\\2\end{array}\right)=10.$$In general, if you need to go $n$ steps to the north and $k$ steps to the west, the number of possible ways for that will be$$\left(\begin{array}{c}n+k\\k\end{array}\right)=\left(\begin{array}{c}n+k\\n\end{array}\right)=\frac{(n+k)!}{n!k!}.$$ 
A: 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 $3\cdot 10=30$.
A: Let us take a generalized case. Say, you have $m$ horizontal roads and $n$ vertical roads from your starting point to your destination. Then you have $n-1$ steps in the horizontal and $n-1$ steps in the vertical. In total you have $(m+n-2)$ steps and lets say you have to choose $(m-1)$ or $(n-1)$ steps from it. Therefore you have a total of $$ \binom{m+n-2} {m-1}$$ ways. Apply it twice once from origin to Starbucks and then to destination.
A: Different way of looking at the question:
Lets call the unit vector towards the west direction $i$ and the unit vector towards the north direction $j$. Then we split the problem into two parts:


*

*Getting to the coffee shop from Central Station :
We have $ai + bj = 1i +2j = i + j + j$ where the vectors could be added in any order, then we ask how many distinct ways could we add these vectors together which is equivelant to the number of distinct permutations of the set $(i,j,j)$, which is ${3 \choose 2}$

*Getting to Times Square from the coffee shop:
By a similar argument to the above our goal is to get the number of distinct permutations of the set $(i,i,j,j,j)$ which is ${5 \choose 2}$ 
Finally to get from Central Station to Times Square we have ${3 \choose 2}{5 \choose 2}=3*10=30$ ways
