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How many pathways?

  1. Feb 15, 2012 #1
    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. jcsd
  3. Feb 15, 2012 #2
    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...
    [tex]\displaystyle \binom{14}{7} = \frac{14!}{7!7!}[/tex]

    For a generic nxn board it would be...

    [tex]\displaystyle \binom{2n-2}{n-1} = \frac{(2n-2)!}{(n-1)!(n-1)!}[/tex]
     
    Last edited: Feb 15, 2012
  4. Feb 15, 2012 #3
    never mind I didnt see the edit.
     
  5. Feb 15, 2012 #4
    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.
     
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