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.

 

[CellCoree]

for your question. The number of paths on a 5x5 chessboard can be calculated using a combination of mathematical principles and logic. First, we can visualize the chessboard as a grid of 25 squares. Then, we can see that in order to reach the upper right corner from the lower left corner, we must make a total of 8 moves (5 moves to the right and 3 moves up).

Next, we can use the concept of combinations to determine the number of ways we can make these 8 moves. This can be calculated as 8!/(5!3!), which is equal to 56. This means that there are 56 different paths that can be taken to reach the upper right corner.

To further understand this calculation, imagine labeling each move as either "R" for right or "U" for up. Then, the number of possible paths can be represented as the number of different combinations of 5 "R"s and 3 "U"s, which is equal to 56.

I hope this helps answer your question and provides some insight into how we can approach solving this type of problem. Please let me know if you have any further questions.