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

Question

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 $x_i$?

Answer 1

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.