If a question asked: How many integer solutions of $x_1+x_2+x_3+x_4=30$ with $-9 \leq x_i \leq 21$? How would this be solved?

I understand how to solve if the inequality was $0 \leq x_i \leq 21$?, but how to solve between a negative and positive inequality?

$N = \binom{30 + 4-1}{30}$

$N(A_i) = \binom{(30 - ?) +4-1}{30 - ?}$


  • $\begingroup$ It is the coefficient of $x^{30}$ in $$\frac{x^{-36}(1-x^{30})^4}{(1-x)^4}$$ $\endgroup$ – Matt Samuel Dec 10 '18 at 20:44
  • $\begingroup$ @MattSamuel Could you explain this? We have to use sets (finished generating functions chapter). $\endgroup$ – Violet Jung Dec 10 '18 at 20:46
  • $\begingroup$ The formula for $N$ you give is the unrestricted value. That is, it would be the answer if the question was how many solutions are there to $x_1+x_2+x_3+x_4=30$ with $x_i≥0$. More work is needed to handle a cap, as in $0≤x_i≤21$. $\endgroup$ – lulu Dec 10 '18 at 20:58
  • $\begingroup$ "set theory" has nothing to do with your question. $\endgroup$ – Jean Marie Dec 10 '18 at 21:47
  • $\begingroup$ @JeanMarie Ah, sorry, I assumed so because that's our current chapter. $\endgroup$ – Violet Jung Dec 10 '18 at 22:34

Let $y_i=x_i+9$. Then $$ y_1+y_2+y_3+y_4=66\\ 0\leq y_i\leq 30 $$ Now find the number of solutions the way you say you know.


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