Jtdylktuy

$x_1 + x_2 + x_3 = 10, 0 leq x_1 leq 10 , 0 leq x_2 leq 6 , 0 leq x_3 leq 2 $

$[x^{10}] (1 + x^1 + x^2 + ... + x^{10})(1 + x^1 + x^2 + ... + x^{6})(1 + x + x^2)$

$[x^{10}] Large (frac{1 - x^{11}}{1-x})(frac{1 - x^{7}}{1-x})(frac{1 - x^{3}}{1-x})$

$[x^{10}] Large( frac{(1 - x^7 - x^{11} + x^{18}) (1-x^3)}{(1-x)^3})$

$[x^{10}] Large( frac{1 - x^3 - x^7 + x^{10}}{(1-x)^3})$

$[x^{10}] (1 - x^3 - x^7 + x^{10})(1-x)^{-3}$

how to proceed further?

Using
$$(1-x)^{-n}=1+nx+frac{n(n+1)}{2}x^2+frac{n(n+1)(n+2)}{6}x^3+dotsb+frac{n(n+1)dotsb(n+k-1)}{k!}x^k+dotsb$$
We get
$$(1-x)^{-3}=1+3x+frac{12}{2}x^2+frac{60}{6}x^3+dotsb +frac{3cdot 4cdot dotsb (3+k-1)}{k!}x^k+dotsb$$
Now you want coefficient of $x^{10}$ in $(1 - x^3 - x^7 + x^{10})color{blue}{(1-x)^{-3}}$. This can be obtained by collecting coefficient of $x^{10}$ obtained by multiplying the various terms. What I mean by that is we will obtain $x^{10}$ by multiplying $1$ with $color{blue}{x^{10}}$, $x^3$ with $color{blue}{x^7}$, $x^7$ with $color{blue}{x^3}$ and finally $x^{10}$ with $color{blue}{1}$. Thus the coefficient of $x^{10}$ obtained will be
begin{align*}
&=(1)color{blue}{left(frac{3cdot 4 dotsb 12}{10!}right)}+(-1)color{blue}{left(frac{3cdot 4 dotsb 9}{7!}right)}+(-1)color{blue}{left(frac{3cdot 4 cdot 5}{3!}right)}+(1)color{blue}{(1)}\
&=66-36-10+1\
&=21.
end{align*}