Combinatorics - falling ball through a labyrinth 
The ball falls from point A to point B. I have some difficulty understanding the concept of what seems to be a very basic combinatorial  problem.
a) You can count the ways, the answer is 6.   But what is the correct mathematical notation here? 
... $2+2+2$ or $3*2$ or something else?
Now here is my thinking:

a) There are 3 layers (3 colors) where the ball can make turn ...
b) There are 5 layers (5 colors) where the ball can make turn ...
How do I apply mathematical notation here? What is the thinking process in these two cases? 
 A: A path by which the ball can reach the bottom can be written as a sequence of $R$ and $L$, with $R$ indicating a movement down to the right, and $L$ a movement down to the left. For example, the path $RLLRLR$ is shown below:

In this labyrinth, the ball visits exactly 6 intersections on the way to the bottom.  The requirement that the ball end in the middle at $B$ means that the number of $R$'s and $L$'s must be equal.  So any path consists of exactly 3 $R$'s and 3 $L$'s, and we want to know how many such paths there are.
This is equivalent to the number of ways of choosing 3 objects out of a group of 6: we have 6 levels of intersections through which the ball passes, and in selecting a path we must select exactly 3 of these levels for the $L$'s, leaving the other 3 as the $R$'s. (In the example above, levels 2,3, and 5 are selected to be $L$.)  In mathematical notation this is $\binom 63,$ which is equal to $$\frac{6!}{3!3!} = \frac{720}{6\cdot 6} = 20.$$
Similarly, in the smaller labyrinth the ball passes through 4 levels of intersections and must choose exactly 2 of these to be $L$, leaving the other two to be $R$, so the answer is $$\binom42 = \frac{4!}{2!2!} = \frac{24}{2\cdot 2} = 6$$
and in general the answer for an $n\times n$ labyrinth is that there are $\binom{2n}{n}=\frac{(2n)!}{n!n!}$ paths to from the top to the bottom corner.
A: Hint: If you rotate the shapes 45$^\circ$ counterclockwise and view the labyrinths as grids, then your problem is the same as counting how many paths are there to move from point A to B with only down and right moves. If your grid is $n\times n$, this is equivalent to counting how many ways can $n$ be expressed as a sum of integers.
A: In an $m*n$ grid, you have $m+n$ slots to fill in to reach to the other corner, m of them will be L (or R) in the notation of soln above. so, fill them up -- how many distinct m slots can you pick out of $m+n$.
