- I want to solve the equation:
$$ x_1 + x_2 +...+ x_n = s \\ \text{where }\ 0 < x_i \le c $$ - Or to rephrase the question count number of ways to break number $s$ in to $n$ numbers where each number is between $1$ to $c$.
- Without upper bound $x_i < c$ I can use stars and bars method. But since i have upper limit can i still use same method with some modification? Or can you reference me to some other method which can help me solve this.
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@Mike Earnest I read your answer in the link. Its really helpful. But i am not sure how the equation will change since i have $xi>0$ and in your answer its $xi≥0$. Just $n$ will be removed from top? – poojan124 Nov 01 '19 at 15:56
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4@poojan124 If you want $x_i \ge 1$, write $x_i = 1 + t_i$ and that becomes $t_i \ge 0$. – Robert Israel Nov 01 '19 at 16:08
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Consider the generating expression that is the coefficient of $x^s$ in $(x+x^2+x^3\dots+x^c)^n$. – Certainly not a dog Nov 01 '19 at 16:08