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GeoGebra

Grid Path Combinations

Author:Hafsa
The Scenario: Imagine you are controlling a robot navigating a city grid starting at Point A(0,0). Your mission is to reach the target at Point B. But there is a catch! You have limited battery, so you must take the shortest path possible.
  • Then you should only move Right (R)or Up (U) .
  • You are only allowed to walk in grey lines. You cannot move diagonally!
  • Therefore, in this model, you are strictly limited to Right and Up moves only.
How to Explore:
  1. Use the SlidersToolbar Image to set the distance of the target (How many steps Right & Up).
  2. Click "Generate another path!" button Toolbar Image to see how the computer creates a random route using only your allowed steps.
Task 2: Draw Your Own Path: I enabled the Pen or PointToolbar Image and Segment Toolbar Image tools for you.
  • Try to draw a some of the shortest paths from A to B(3,4)
  • Check: For every path, you used exactly 3 Right and 4 Up steps, right? (You have to, otherwise you won't reach B!)
The Goal: In a small grid, it may be easy to draw paths. But as the grid gets larger, there may be hundreds of different ways to reach the destination. Our goal is not to draw them all, but to discover the mathematical pattern behind them.
Part 2: Unlocking the Formula Look at the generated paths again. Even though every path looks different on the map, they all share a secret:
  • To reach a target at Right = r and Up = u, you strictly need r Right moves and u Up moves.
  • Total Steps (t) = r + u
  • Every path is just a scrambled "word" made of these letters.
    • Example for (3,4): RRRUUUU, URURURU, UUUURRR...
Instead of drawing lines, think about empty boxes for your total steps: [ _ ] [ _ ] [ _ ] [ _ ] [ _ ] [ _ ] [ _ ] You have 7 total steps. You just need to choose which 3 of them will be "Right" (R). Once you place the R's, the U's automatically fill the empty spots! So, the question is simply: "In how many ways can we choose 3 spots out of 7?" This is the definition of Combination:        Total number of Paths = == (See t - r = u) Warning: You might ask: "Why did we choose Right moves? Can't we choose Up moves?" Yes, you can!
  • Selecting 3 spots for Right is exactly the same as leaving 4 spots for Up.
  • Mathematically, choosing r items is the same as choosing the remaining (n-r) items.
  • Remember the property:=

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