Find out the number of solutions to the equation- $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 25$ under certain constraints How many solutions are there to the equation $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 25$ where each $x_i$ is a non-negative integer, $3 \leq x_1 \leq 10, \ 2 \leq x_2 \leq 7$ and $x_3 \geq 5$
I have been able to do all my counting problems but this one. I can not find an equation for max number per variety.
I know that I will subtract the min for each $x$ from the total, but I can not do $\begin{pmatrix} 15 \\ 6\end{pmatrix}$ cause that would not factor in the max per variety. 
What is the equation I would use for this?

The answer will be the coefficient of $x^{25}$ from the below expression-
$$(x^3+x^4+ \dots+x^{10})(x^2+x^3+ \dots+x^7)(x^5+x^6+ \dots+x^{20})(x^0+x^1+ \dots+x^{15})(x^0+x^1+ \dots+x^{15})(x^0+x^1+ \dots+x^{15}).$$
