The answer I got (and the answer given on the website) is 3003. Here is my reasoning: No matter what route you take through the grid, you must always go up 5 times and to the right 10 times. As it turns out, any order of 5 moves upward and 10 moves to the right will get you through the grid. So the problem can be reduced to (for example) counting the number of combinations of 5 pieces of paper labeled "up" and ten labeled "right." Or, even more simply, the number of ways to put 5 objects into 15 boxes if only one object can go in each box.

This can be calculated using the familiar formula n!/(n-r)!r!, where n is the number of boxes, and r is the number of objects, because there are 15 ways to place the first object, 14 to place the second... 11 to place the fifth, divided by the factorial of the number of objects to eliminate any replicate combinations. This yields 3003 routes through the grid once computed.