Counting Paths – Grid Routes and Combinations
Imagine standing at the top-left corner of a grid, allowed to move only right or down at each step, trying to reach a point some number of blocks right and some number of blocks down. Every valid path uses exactly the same total number of moves – the only thing that changes between paths is the order in which the right-moves and down-moves happen.
This connects directly back to combinations: if a path needs a right-moves and b down-moves, counting the paths is the same as choosing which a of the total a + b moves are the right-moves – giving the formula C(a + b, a).
Counting Grid Paths
A path needing a right-moves and b down-moves has C(a + b, a) possible routes – the number of ways to choose which moves are “right” among all a + b total moves.
C(2 + 3, 2) = C(5, 2) = 10 paths.
Real-Life Application
- City navigation: counting possible routes through a grid-based street layout.
- Robotics: counting possible movement paths for a grid-based robot.
- Game design: counting possible movement paths in grid-based board games.
Key Takeaways
- A grid path from corner to corner uses a fixed total number of right- and down-moves.
- Counting paths is equivalent to choosing which moves are “right” among the total.
- The formula is C(a + b, a), a direct application of combinations.
Practice: Counting Paths
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