Compute the number of positive integers solutions of the following equation:
$x_{1}+2x_{2}+3x_{3}+10x_{4}+2x_{5}=n,$
where $x_{1}\le 4$, and $3\le x_{2}\le 7$.
I am using the generating functions approach and have come up with
$\frac{1-x^5}{1-x} \times\frac{1-x^8}{1-x^4}\times\frac{1}{1-x^3}\times\frac{1}{1-x^10}\times\frac{1}{1-x^2}$
Given the boundaries I am unsure if the generating functions for $x_{1} and$ $ 2x_{2}$are correct
You have not formulated properly the product.
The product defining the OGF of the number of positive integral solutions is $$ \eqalign{ & \left( {z^{\,1} + z^{\,2} + z^{\,3} + z^{\,4} } \right)\left( {\left( {z^{\,2} } \right)^{\,3} + \cdots + \left( {z^{\,2} } \right)^{\,7} } \right)\left( {\left( {z^{\,3} } \right)^{\,1} + \left( {z^{\,3} } \right)^{\,2} + \cdots } \right) \cdot \cr & \cdot \left( {\left( {z^{\,10} } \right)^{\,1} + \left( {z^{\,10} } \right)^{\,2} + \cdots } \right)\left( {\left( {z^{\,2} } \right)^{\,1} + \left( {z^{\,2} } \right)^{\,2} + \cdots } \right) \cr} $$ If you try to expand it, you will see that the construction of the coefficient of $z^n$ obeys to the given system.
hint: start with a small example, to understand how to fit the coefficients of the linear combination plus the boundaries