Number of Paths on 5x5 Chessboard: Solve the Puzzle!

[westgrant88]

I know there is a formula for how many of a certain thing like squares on a 8x8 chessboard but I haven't come across 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

 

[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...

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

 

[westgrant88]

never mind I didnt see the edit.

 

[Number Nine]

Just wondering how you get 7 moves right and 7 up. Icount 4 and 4. How am I missing 3?

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.