## Up or Right

Starting at the point $P(x,y)$ on the coordinate plane, a pin can be moved either to point $A(x+1,y)$ or to point $B(x,y+1)$. If the pin starts at $(0,0)$ and is moved to $(4,4)$, what is the probability that it passed through $(2,2)$?
Source: NCTM Mathematics Teacher, February 2006

SOLUTION
Under the given constraint the pin can only move either one unit up or one unit right at a time. The following figure shows there is $1$ path from $(0,0)$ to $(0,1)$ and $1$ path from $(0,0)$ to $(1,0)$. We write a $1$ next to the points to indicate the number of path leading to the them

Likewise, there is $1$ path from $(0,0)$ to each of the other points on the vertical and horizontal axes

The number of paths from $(0,0)$ to $(1,1)=1+1=2$

The number of paths from $(0,0)$ to $(2,1)=2+1=3$

If we keep working this way, we find that there are $70$ paths from $(0,0)$ to $(4,4)$

There are $6$ paths from $(0,0)$ to $(2,2)$ and by the same token $6$ paths from $(2,2)$ to $(4,4)$. The number of paths from $(0,0)$ to $(4,4)$ passing through $(2,2)$ equals
$6\times 6=36$
Probability of pin passing through $(2,2)$ on the way from $(0,0)$ to $(4,4)$  equals
$36/70=18/35$

Answer: $18/35$