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This is the correct answer. Congrats, @Scroogey. I wonder if you'd be interested in taking a look at these bitcoin math puzzles i've been doing too.
Also, for this question, I was hoping someone would solve it by drawing out the grid and writing the number of paths from each grid point, essentially working backwards recursively from the lower right to the upper left. There's an interesting connection between C(n,r) and triangular sums.
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That feels right to me
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What the 6 routes for the 2x2 grid have in common is that they all consist of exactly 2 right and 2 down moves, for 4 moves total.
This pattern holds for any size grid: there must be 20 steps for the 10x10 grid, exactly 10 down and 10 right.
How many combinations are there for 10 down and 10 right?
with n=20,r=10 gives 184756.