How many 5 block paths can take you home if walking 2 blocks north then 3 blocks west got you here? This should feel familiar if you're from NYC, Chicago, Barcelona, or Toronto!
If I were to travel only north and east, how many routes did I have to get to the castle?
The diagram above shows a tiling of 7 squares with \(A\) and \(B\) denoting the initial point and destination point respectively. The arrow signs indicate the direction that you're able to travel to (east, south, north-east). Find the total number of ways to reach point \(B\) from point \(A\).
Bonus: Generalize this for \(n\) squares.
There are 34 people in line to get into a theatre. Admission costs 50 cents. Of the 34 people, 28 of them have a 50-cent piece and 6 have a $1 bill. Suppose that the box office has an empty cash register, how many ways can the people line up so that change is available when needed?
Note: For this question, assume that the people are indistinguishable.
The rook is a chess piece that can only move horizontally or vertically. There is a rook placed on an 8 by 8 empty chessboard on the top left hand corner. It is to arrive at the bottom right hand corner. If it can only move downwards or right, and can only travel 1 space per turn, how many different ways are there to accomplish this task?