<s>[INST] Homework - How many non negative solutions? $a+b+c+d+e = 30$
we know that $10\leq e$ and that $4\leq d \leq 7$
How many non negative solutions does this equation have?
 [/INST] That would be the coefficient of $x^{30}$ in 
$$P(x)=(1+x+x^2+...)^3(x^4+x^5+x^6+x^7)(x^{10}+x^{11}+x^{12}+...)$$
Noting we can write $P(x)$ as$\dfrac{x^{14}}{(1-x)^4}(1+x+x^2+x^3) $, we need the sum of coefficients of $x^{13}, x^{14}, x^{15}, x^{16}$ in  $\dfrac1{(1-x)^4}$.  Use the Binomial Theorem for finding these coefficients.  
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