Walking in a Rectangular Grid Warmup

Probability Level 2

As a tourist in NY, I want to go from the Grand Central Station $(42^\text{nd}$ street and $4^\text{th}$ avenue $)$ to Times Square $(47^\text{th}$ street and $7^\text{th}$ avenue $).$

If I only walk West and North, how many ways are there for me to get there?

15 35 45 56

1 solution

Brilliant Mathematics Staff
Aug 1, 2020

Solution 1: We will be walking 8 blocks, of which 3 are west and 5 are north. Thus, the number of ways is ${ 8 \choose 3 } = 56$ .

Solution 2: Starting from the lower right, we label the number of ways to get to a particular intersection by just going north or west. This can be obtained by finding the sum of the number of ways from an intersection that is south or east of it.

Hence, the answer is 56.

Can you elaborate on the first, solution. I have never heard that before. Second is the standard approach, nice!

Mahdi Raza - 10 months ago

Hi Mahdi, please check out rectangular grid walk .

Brilliant Mathematics Staff - 10 months ago

Ok, thanks for sharing!

Mahdi Raza - 10 months ago

Elaborating on the first solution. What we have is that we must go 5 blocks up and 3 blocks left. Each path is just a different arrangement of the 8 movements. Therefore, there are eight choose three ways to arrange them. Why is this? The addition is just a Pascal's triangle, which is made up of different X choose Y combinations. You can solve it by assuming it is a tilted Pascal's triangle.

Rick Zhou - 6 months, 3 weeks ago

Walking in a Rectangular Grid Warmup

1 solution

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