<s>[INST] What is the coefficient of the term $x^4 y^5$ in $(x+y+2)^{12}$? What is the coefficient of the term $x^4 y^5$ in $(x+y+2)^{12}$?
How can we calculate this expression  ?
I've applied the binomial theorem formula and got $91$ terms but I am not sure if it is right or wrong.
 [/INST] The trick is to apply trinomial formula here, the coefficient is
$${12\choose4,5,3}2^3=\frac{12!2^3}{4!5!3!}$$
To see this, first consider $(x+y)$ as a whole body and then use binomial formula. Then the coefficient of term $(x+y)^9$ is 
$\displaystyle{12\choose 9}2^3=\frac{12!2^3}{9!3!}$ (Note we don't need to consider terms like $(x+y)^k$ other than $k=9$ here). And then consider the term $x^4y^5$ in $(x+y)^9$, whose coefficient shall be $\displaystyle{9\choose4}=\frac{9!}{4!5!}$. Combine two things, we obtain
$${12\choose 9}2^3{9\choose4}=\frac{12!2^3}{9!3!}\frac{9!}{4!5!}=\frac{12!2^3}{4!5!3!}$$
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