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How many nonnegative integer solutions are there to the equation $x_1+x_2+x_3+x_4 = 18$ with $$−2\leq x_3 \leq 4\\0 \leq x_2 \leq 5\\3 \leq x_1 \leq8\\ −3 \leq x_4 \leq 13$$
when its in is simple form its easy to solve that but here in this kind of question how i solve it ?
I never miss a chance to recommend a good book. "How to Count" covers the relevant material and even more and it's a good book indeed.
Now to the problem. Assuming mistranslating "nonnegative" has a higher likelihood than copy and pasting math expressions, twice (one for "$-2 \leq ...$" and one for "$-3 \leq ...$") I will stick to the following version:
How many integer solutions are there to the equation ...
This class of problems is solved with what is called generating functions. First of all $$−2\leq x_3 \leq 4 \Leftrightarrow 0 \leq x_3+2 \leq 6$$ $$−3\leq x_4 \leq 13 \Leftrightarrow 0 \leq x_4+3 \leq 16$$ So, with a substitution like $y_1=x_1, y_2=x_2, y_3=x_3+2, y_4=x_4+3$ the problem is equivalent to $$y_1+y_2+y_3+y_4=23$$ $$3\leq y_1 \leq 8$$ $$0\leq y_2 \leq 5$$ $$0\leq y_3 \leq 6$$ $$0\leq y_4 \leq 16$$
The generating function for this problem is $$(y^3+y^4+...+y^8)(y^0+y^1+...+y^5)(y^0+y^1+...+y^6)(y^0+y^1+...+y^{16})$$ the coefficient near the $y^{23}$ term is the answer, which is $232$.
