<s>[INST] generating functions / combinatorics Calculate number of solutions of the following equations:
$$ x_1 + x_2 + x_3 + x_4 = 15 $$
where $ 0 \le x_i < i + 4 $ 
I try to solve it using generating functions/enumerators :
$$ (1+x+x^2+x^3+x^4)(1+x+x^2+x^3+x^4+x^5)(1+x+x^2+x^3+x^4+x^5+x^6)(1+x+x^2+x^3+x^4+x^5+x^6+x^7)$$
and take coefficient near $15$. But I do not know how to quickly calculate it. Maybe there exists any faster way?
 [/INST] Use the stars and bars technique to solve for the number of solutions without restrictions, and look at how to count the solutions that violate the restrictions.
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