Questions $41-45$ is a guided tour to finding the number Cn of valid paths from $(0,0)$

to $($n $+1,$ n $+1).$

Note: you can use the result from Question k to answer Question $($k $+1),($k $=41-$

$44).$

A path P from $(0,0)$ to $($n $+1,$ n $+1)$ is valid if the following conditions are met:

i$.$ Every valid path P starts from $(0,0)$ and stops on reaching $($n $+1,$ n $+1).$

ii$.$ If $($a$,$ b$)$ is a point on a valid path P then exactly one of $($a $+1,$ b$)$ or $($a$,$ b $+1)$

is a point on the valid path P$.$

iii. Each valid path P constitute exactly of points $($a$,$ b$)(0$ a$,$ b n $+1)$ as

described by the rules i$.$ and ii$..$

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