How to find the coefficient of $x^2y^3z^6$ in the expression $(x^2+y+z)^{10}$? How can find the coefficient of $x^2y^3z^6$  in the expression $(x^2+y+z)^{10}$? 
I know this involves the binomial theorem, but I can't figure out how to solve it. 

We can apply the binomial theorem twice in order to find the coeffcient. It is convenient to use the coefficient of operator $[x^k]$ to denote the coefficient of $x^k$ of an expression.

We obtain
  \begin{align*}
[x^2y^3z^6](x^2+y+z)^{10}&=[x^2y^3z^6]\sum_{k=0}^{10}\binom{10}{k}x^{2k}(y+z)^{10-k}\tag{1}\\
&=[y^3z^6]\binom{10}{1}(y+z)^9\tag{2}\\
&=10[y^3z^6]\sum_{k=0}^9\binom{9}{k}y^kz^{9-k}\tag{3}\\
&=10\binom{9}{3}\tag{4}\\
&=840
\end{align*}

Comment:


*

*In (1) we apply the binomial theorem once to $(x^2+(y+z))^{10}$.

*In (2) we select the coefficient of $x^2$ which is given when the index $k=1$.

*In (3) apply the binomial theorem again.

*In (4) we select the coefficient of $y^3z^6$ corresponding to the index $k=3$.
