Calculating Rabbit's Moves on a Chessboard | Combinatorial Question

[ParisSpart]

A rabbit lives on an ordinary chessboard 8x8. The rabbit located over the cell (1.1) (lower left, the upper right corner is (8.8), the lower right corner is (8,1)). At any time the rabbit jumps either to the right or upwards.

How many different ways can move the rabbit to reach the cell (3.4)?


I have a problem here.I think all the moves in the chessboard is (64 C 8) like (n C k) combinations, but i can not think how to move from the cell (1.1) to (3.4) maybe someone can give me any suggestion how?

 

[Simon Bridge]

To understand the problem - draw out the chessboard and locate the cells.
Start tracing out possible paths manually - at some point you'll see a pattern emerge.

 

[ParisSpart]

but the there are a lot of paths... maybe we can find another way to find them with combinatorial types?

 

[Simon Bridge]

Rabbit can only move right or upwards though - that severely restricts the number of paths.

Of course there is a shortcut involving combinations etc.
Your problem is how to find it - with the first step being to understand the problem.

Another way: since you already solved how to get to (8,8) from (1,1) ... presumably you can do it for any size square board?
So how does that work?

 

[ParisSpart]

i will try to estimate it and i will post the result here to find out if its correct

 

[ParisSpart]

i find it there are 10 ways to move right and up in the chessboard!

 

[Simon Bridge]

well done - how did you do that?

 

[ssd]

The rabbit reaches (8,8) from (1,1) in ¹⁴C₇= 14!/(7!.7!) ways, if the rabbit jumps only one square at a time.
Call a rightward jump 'R' and upward jump 'U'. The answer is the total number of linear permutation of 7 R's and 7 U's.
Similarly, to reach (m,n) the rabbit can move in
(m+n-2)!/ [(m-1)!(n-1)!] ways.

 

[Simon Bridge]

Well done.
The answer is not so important as how you got to it.
Also I hope you can see that you sometimes have to mess about with a problem for a bit before finding out the short, simple, way to do it. Don't be scared to start a problem without knowing how to get to the answer.