# How many pathways?

1. Feb 15, 2012

### westgrant88

I know there is a formula for how many of a certain thing like squares on a 8x8 chessboard but I havent come accross anything on pathways .Is there one for this type of problem?Any help would be appreciated.
Consider a 5-by-5 chessboard. You want to move a nickel from the lower left corner to the upper right corner. You are only allowed to move the nickel one square at a time, and each move must be either to the right or up. How many different paths are possible?
Thanks

2. Feb 15, 2012

### Number Nine

For an 8x8 board...
If you can only move up or right, then that simplifies things considerably. Note that no matter which path you choose, you must move right 7 times and up 7 times, so we have 14 "moves" in total, no matter which path you choose. Basically, we have 14 slots and want to know how many ways there are to insert 7 ups and 7 rights into those slots. Really, we just want to know the number of ways to arrange 7 objects in 15 slots, since we then get the other 7 for free (just use the empty slots). The solution is then...
$$\displaystyle \binom{14}{7} = \frac{14!}{7!7!}$$

For a generic nxn board it would be...

$$\displaystyle \binom{2n-2}{n-1} = \frac{(2n-2)!}{(n-1)!(n-1)!}$$

Last edited: Feb 15, 2012
3. Feb 15, 2012

### westgrant88

never mind I didnt see the edit.

4. Feb 15, 2012

### Number Nine

That's my bad; I did it for an 8x8. For a 5x5, you're right, it would be 4 moves up and 4 moves right.