How to compute the coefficient of an equation? What is the coefficient of $x^2y^2z^3$ in $(x + 2 y + z)^7 $?
This is the question at a test and the correct answer is given as 840.
Isn't it $7!/(2!2!3!)$ ?
 A: The coefficient of $x^2y^2z^3$ in $(x+y+z)^7$ is indeed $\frac{7!}{2!\cdot2!\cdot3!}=210$
however here we have $2y$ instead of $y$. So instead of $y^2$ we have $4y^2$. Multiplying your result by $4$ gives the desired result.
A: $\newcommand{\+}{^{\dagger}}
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\begin{align}
\pars{x + 2y + z}^{7}&
&=\sum_{{\vphantom{\LARGE A}a,b,c=0}\atop
        {\vphantom{\huge A}a + b + c = 7}}^{\infty}
{7! \over a!\,b!\,c!}\,x^{a}\pars{2y}^{b}z^{c}
=\sum_{{\vphantom{\LARGE A}a,b,c=0}\atop
        {\vphantom{\huge A}a + b + c = 7}}^{\infty}
{7! \over a!\,b!\,c!}\,2^{b}\pars{x^{a}y^{b}z^{c}}
\end{align}

$$
{7! \over 2!\,2!\,3!}\,2^2 = 7\cdot 6\cdot 5\cdot 4 = \color{#66f}{\large 840}
$$

