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Number of Paths Using Permutations

  1. Dec 1, 2012 #1
    I'm pretty sure I'm right, but I'd appreciate it if I could obtain some verification.

    1. The problem statement, all variables and given/known data
    Consider the following grid:
    2ij0xw6.jpg

    The goal is to move from point [itex]\alpha[/itex] to point [itex]\beta[/itex] to point [itex]\gamma[/itex] by moving along the edges of the grid from point to point. You can only move to the right or down from [itex]\alpha[/itex] to [itex]\beta[/itex], and you can only move left or up from [itex]\beta[/itex] to [itex]\gamma[/itex]. You cannot move outside the grid.

    How many distinct paths are there from [itex]\alpha[/itex] to [itex]\beta[/itex] to [itex]\gamma[/itex]?

    2. The attempt at a solution
    I split this into two parts. The first part, [itex]\alpha[/itex] to [itex]\beta[/itex], consists of some combination of moves, but always consists of moving down 4 times and to the right 4 times. Thus, the total number of paths will be [itex]\frac{8!}{4! \cdot 4!} = 70[/itex].

    The second part is from [itex]\beta[/itex] to [itex]\gamma[/itex]. It's the same thing, but with 2 moves up and 2 moves left. Thus, the number of paths will be [itex]\frac{4!}{2! \cdot 2!} = 6[/itex].

    Thus, the total number of paths is [itex]70 \cdot 6 = 420[/itex]. Am I correct?
     
  2. jcsd
  3. Dec 1, 2012 #2

    Dick

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    Seems fine to me.
     
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