MHB Counting Routes in a 4x6 Grid: How Many Ways to Reach the End?

[Jameson]

Consider a 4x6 grid. You begin in the bottom left corner and want to navigate to the top right corner. You can only move right or up, and you can move just one space per move. How many ways are there to get to the end point?

There are at least two ways of solving this. One uses a method of counting arrangements of a set and the other involves looking at each point as a sub-destination that can be reached in a finite number of ways as well.
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[Jameson]

Congratulations to the following members for their correct solutions:

1) MarkFL
2) Sudharaka

Solution (from MarkFL): [sp]Every path from start to end contains 6 moves to the right and 4 moves up, for a total of 10 moves. Thus, to find the number of unique paths, we simple find the number of ways to choose 6 from 10 or equivalently 4 from 10. So, the number $N$ of unique paths is given by:

$\displaystyle N={10 \choose 6}={10 \choose 4}=210$[/sp]