Going to the castle - Part 1

Probability Level 1

I was on vacation in England, and wanted to visit the Tower of London. The roads were laid out on a grid map, and the castle was 4 blocks north and 5 blocks east. If I were to travel only north and east, how many routes did I have to get to the castle?

9 \choose 4

5 \choose 4

7 \choose 4

11 \choose 4

5 solutions

Hannah Ford
May 9, 2015

Imagine your route as a list of 4Ns and 5Es. There will be 9 letters in total (detailing roads taken), and you need to choose which 4 are North. Hence 9 C 4. (You could have alternatively selected the 5 roads that would be East, giving 9 C 5 as a second possible solution).

That's great!

Chung Kevin - 6 years ago

why do you use only 4 why not any number else , i know there will be a specific reason , it will be highly appreciable if you can explain it .

Syed Hissaan - 4 years, 4 months ago

Ryan L.
Oct 16, 2016

Here is another way of thinking about this problem.

You have four $N$ 's and five $E$ 's. Thus the total number of ways to arrange the $N$ 's and $E$ 's is $9!$ . But since our $N$ 's and $E$ 's are indistinguishable we must divide out the number of ways to permute our $N$ 's and $E$ 's. Thus our answer is $\frac{9!}{4!5!}$ ... or $9\choose 4$ .

John Wang
May 20, 2015

You can also use the pascal's triangle. If you count the ways in a 1x1 grid, there are 2 ways, since the upper right corner is like the 2 in a pascals triangle. In a 2x2 grid, there are 6 ways, since the upper right corner is the 6 in a pascal's triangle.

Alan T
Jan 1, 2018

Or its permutation with repetition.

Thor Stead
Dec 8, 2015

Hannah Ford's way is exactly how I did it :)

Going to the castle - Part 1

5 solutions

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