Going to the Castle - Part 2

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 and the castle was 4 blocks north and 5 blocks east. Unfortunately, there was some police activity near the hotel, and that intersection was blocked off so I could not go through it.

If I were to travel only north and east, how many routes did I have to get to the castle?

{ 3 \choose 1 } { 6 \choose 2 }

126 - { 3 \choose 1 } { 6 \choose 2 }

{ 3 \choose 2 } { 6 \choose 1 }

126 - { 3 \choose 2 } { 6 \choose 1}

2 solutions

Rohit Sachdeva
May 9, 2015

The tower is at (4N,5E). The total number of ways to reach the tower is arrangement of 4N's & 5E's which can be done in $^{9}C_{4}$ ways

Now the police is at (2N,1E). The total number of ways to reach the police is arrangement of 2N's & 1E's which can be done in $^{3}C_{2}$ ways

After reaching the police, tower is at (2N,4E). The total number of ways to reach the tower from police is arrangement of 2N's & 4E's which can be done in $^{6}C_{2}$ ways.

So required number of ways to reach the tower without reaching the police is $^{9}C_{4} - (^{3}C_{2} * ^{6}C_{2}) = 126 - (^{3}C_{1} * ^{6}C_{2})$

Good solution !! :)

jaiveer shekhawat - 6 years, 1 month ago

Isn't 126 - (3C2 * 6C2) = 126 (3C1 * 6C2) ??

Terza Reyhan - 6 years ago

3C2=3C1=3!/((3-2)! 2!)=3!/((3-1)! 1!) There's no reason to change the equation any further

Nacho Diaz - 3 years, 7 months ago

Brody Burkett
May 30, 2018

There are 9C4 ways to traverse the grid without restriction. However you have to add the number of ways to the police (restricted) and the number of ways to the tower from the police (implicitly restricted). This is 3 × (6C2) = 45. Subtract this total number of illegal paths from 126.

Going to the Castle - Part 2

2 solutions

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