# Number of integral soultions to linear equations without unit coefficients

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 $$\sum_{N=0}^\infty s(N) z^N = \prod_{i=1}^k \dfrac{1}{1 - z^{a_i}}$$
• So, if I have a linear equation like $2*(x_{1}+x_{2}+\cdots+x_{6})+x_{7} = C$ , (C being some integer) then the right hand side of the above equation becomes : $\prod_{k=1}^7 \frac{1}{1-a_{i}}$ = $(1-x^{2})^{-6}(1-x)^{-1} = (1-x)^{-7} (1+x)^{-6}$. Suppose I have to extract the answer for S(10), then should I find the coefficient of $x^{10}$ in the right hand side for my answer? – nerdier.js Sep 5 '14 at 18:43