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Consider a coin that can move in at most two ways when it is placed on any square of the usual (8 x 8) chessboard (64 squares). From its current square it can move either to the adjacent square above (if it is not in the top most row) or to the adjacent square on the right (if it is not in the rightmost column).

What is the number of distinct paths in which the coin can move from the lower left corner square to to the top right corner square ?

I understand the problem and it should be solved using recursion technique but every time I approach this kind of problem I take lot of time. Sometimes I get the same problem in different flavor which says it can only move down or left(any combination of left/right/up/down).

Can someone give me some ideas how to start with these problem?

I found answer to be 688 which may be wrong.

Help appreciated :)

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    $\begingroup$ Please add your working and explanation of how you got your answer of 688. This will allow others to better understand your thoughts on the problem and hence give meaningful suggestions. $\endgroup$ – Ian Miller Nov 22 '16 at 6:06
  • $\begingroup$ For a $2$*$2$ - 2 ways $3$*$3$ - It consists of a top right $2*2$ square and and it can go two positions right and two positions top. so for a $3$*$3$ it will be $2*2 + 2=6$ $\endgroup$ – ViX28 Nov 22 '16 at 6:10
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At the and we have seven times $\rightarrow$ and seven times $\uparrow$.

For example;

$$\rightarrow \rightarrow \uparrow \rightarrow \uparrow \rightarrow\rightarrow\rightarrow \uparrow\uparrow\uparrow \rightarrow \uparrow\uparrow$$

It define a path. Hence all possible paths: $$14!\over {7!.7!}$$

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  • $\begingroup$ Thanks a lot :) I was thinking there would be some nicer ways to solve this and u gave it :) $\endgroup$ – ViX28 Nov 22 '16 at 6:25
  • $\begingroup$ @ViX28: You are welcome :) $\endgroup$ – mesel Nov 22 '16 at 6:25

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