Starting at the point on the coordinate plane, a pin can be moved either to point or to point . If the pin starts at and is moved to , what is the probability that it passed through ?

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 path from to and path from to . We write a next to the points to indicate the number of path leading to the them

Likewise, there is path from to each of the other points on the vertical and horizontal axes

The number of paths from to

The number of paths from to

If we keep working this way, we find that there are paths from to

There are paths from to and by the same token paths from to . The number of paths from to passing through equals

Probability of pin passing through on the way from to equals

**Answer**: