Lattice paths
The question
A counter crosses a board 7 blocks wide and 2 blocks tall, moving only right or up one block at a time from the bottom-left corner to the top-right corner.
How many different routes are there?
Count the routes without drawing them
wide:7 tall:2
7+2=9
1×9=9
9÷1=9
9×8=72
72÷2=36
36
- The grid is 7 wide and 2 tall, and you walk only right or up from one corner to the far one.
- Every route is the same length — 7 steps right and 2 up, 7 + 2 = 9 in all. So a route is nothing but a choice of WHICH 2 of those 9 steps go up.
- That is 9 choose 2.
- So 36 routes — not 512, because most free right-or-up choices run off the grid.
How it works.
- 01
The grid is 7 wide and 2 tall, and you walk only right or up from one corner to the far one.
- 02
Every route is the same length — 7 steps right and 2 up, 7 + 2 = 9 in all. So a route is nothing but a choice of WHICH 2 of those 9 steps go up.
- 03
That is 9 choose 2.
- 04
So 36 routes — not 512, because most free right-or-up choices run off the grid.