There is a MxN grid (see picture) of spots. You start from the top left spot and go to the bottom right spot. At any spot, you can only go DOWN or RIGHT.

How many different ways can you go from the top left to the bottom right spots?

There are:
(m+n-2) CHOOSE (m-1)
=
(m+n-2) CHOOSE (n-1)
ways that you can go from the top left to bottom right spots.

[where “CHOOSE” represents the choose function for non-negative integers (http://en.wikipedia.org/wiki/Combination)]

Any path will require (m-1)+(n-1)=m+n-2 total steps.
Any path (which requires n-1 DOWNward steps) is uniquely identified by the spots at which a DOWNward step is taken.
The number of possible paths, then, can be found by CHOOSing which of the total number of steps (m+n-2) will be DOWNward ones (n-1).

Showing that the above expressions are equal is left as an exercise for the reader :)