Find the number of routes The diagram below shows a $4$ rows $\times$ $6$ columns grid. Find the number of ways to travel from $A$ (at the bottom left) to the top right along the grid lines. At every junction point, one can only go right or upwards.

I can setup a tree diagram and then count (slowly) the total number of routes. The question is how to arrive at the elegant answer : $\binom{8}{3}$.
 A: A standard way of looking at this problem is as follows. 
Each step you take is either an $U$ for up or an $R$ for right.
Notice that whatever you do, you'll need to move $5$ times to the right and $3$ times up to get from the bottom left to the top right corner.
So a route corresponds to an $8$ letter long word made of $3$ $U$ letters and $5$ $R$ letters. Also each such word corresponds to a good route.
Therefore it's enough to count these words. And there are exactly $\binom {8}{5} = \binom{8}{3}$ of these, since out of the $8$ places you have to choose $5$ where you put the letter $R$ - or equivalently out of the $8$ places choose the $3$ where you put the letter $U$.
A: This problem is known as Lattice path : https://en.wikipedia.org/wiki/Lattice_path.
We have the result that the number of lattice paths from $(0,0)$ to $(a,b)$ is : $$\binom{a+b}{a}$$
So  here the answer is indeed $\binom{8}{3}$.
You can obtain this result by induction :
The previous step to reach $(a,b)$ is $(a-1,b)$ or $(a,b-1)$ so by hypothesis of induction and by using some properties of binomial coefficients you can prove the result... (Tell me if you encounter some problemsin the proof by induction).
