To determine the number of integral solutions for the linear equation
$$ x_1+x_2+x_3+\cdots+x_k = N$$
we have an expression $$ ^{N+k-1}C_{k-1}$$
But I want to know if the coefficients of $x_{1}+x_{2}+x_{3}$ were not unity, i.e. it were of type
$$ a_1 x_1+a_2 x_2+a_3 x_3+\cdots+a_k x_k = N$$
then how can we determine the number of integral solutions $>0 $ to this equation? How do we work towards the solution for this?
P.S. I am really not aware if any other question exists as the duplicate of this. Please pardon me if a question exactly like this exists.
The number $s(N)$ of solutions of your equation has generating function $$ \sum_{N=0}^\infty s(N) z^N = \prod_{i=1}^k \dfrac{1}{1 - z^{a_i}}$$