20 × 20 20\times20 Lattice Paths, No Need For A Computer Right?

Probability Level 3

Starting in the top left corner of a 2 × 2 2\times2 grid made out of single 1 × 1 1\times1 squares, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.

How many such routes are there in a 20 × 20 20\times20 grid?

You may use a calculator for the final step of your calculation.

Hint : Pascal's triangle .


The answer is 137846528820.

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L ... by L ... , 21, Austria

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2 solutions

Pranshu Gaba
Aug 26, 2016

This is a direct application of the rectangular grid walk . There is a total of ( 20 + 20 20 ) = ( 40 20 ) = 137846528820 \dbinom{20+20}{20} = \dbinom{40}{20} =\boxed{137846528820} paths from the top-left corner to the bottom-right corner.

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According to the instructions he can only move right and down. so he should take 40 steps to reach from corner to the other corner.Out of them 20 steps should be horizontal or vertical.so total number of ways are 40c20.

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