# How many integer solutions are there to the equation $x_1+x_2+x_3+2x_4+x_5=72$?

If the question given is to find the number of integer solutions to the equation $$x_1+x_2+x_3+x_4+x_5=72$$ where $$x_1\ge2, x_2,x_3\ge1, x_4,x_5\ge0$$

I know that the solution would be:

$$(x_1-2)+(x_2-1)+(x_3-1)+(x_4)+(x_5)=72$$

So, $$x_1+x_2+x_3+x_4+x_5=76$$

And the number of integer solutions would be $${76+5-1 \choose 76}$$

But how would I find the answer if the equation given is $$x_1+x_2+x_3+2x_4+x_5=72$$ with same restrictions on $$x_i$$?

One simple approach would be to assume $$x_4=k$$ and solve the problem for $$x_1+x_2+x_3+x_5=72-2k$$ and sum over all $$k$$.
A more general approach would be to construct ordinary generating functions for each variable in question and find the coefficient of $$x^{72}$$ in their common generating function.