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Difference between non - negative and positive integral solution :

(a) Number of non negative integral solution of equation $x+2y+3z+4w =n$ = Coefficient of $x^n$ in $[(1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1-x^4)^{-1}]$

(b) Number of positive integral solution of equation $x+2y+3z+4w =n$ = Coefficient of $x^{n-(1+2+3+4)}$ in $[(1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1-x^4)^{-1}]$

My question is whether positive integral solution and non negative integral solutions are not same ?

How the power of $x^{n} .....(a)$ varies with $x^{n-(1+2+3+4)}....(b)$ please clarify on this thanks.

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In the second case you can't have $xyzw=0$ i.e. they all have to be at least $1$ and none can be zero.

If you subtract $1$ from each of $x,y,z,x$, and $10$ from $n$ then you transform the problem into the first case.

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