<s>[INST] determine the coefficient of $x^2yz$ in the expansion of $(2x-y+z+1)^7$ 
determine the coefficient of $x^2yz$ in the expansion of $(2x-y+z+1)^7$

I found this question in my textbook , i seem very easy in the beginning but my answer is wrong according to answerkey.
I said that the coefficient is equal to $\frac{7!}{2! \times 1 \times 1 \times 3!} \times 2^2 \times (-1)$
However ,the answer is $-280$. What am i missing ?
 [/INST] A small crosscheck. We use the coefficient of operator $[x^k]$ to denote the coefficient of $x^k$ in a series. This way we can write for instance
\begin{align*}
[x^k](ax+z)^n=\binom{n}{k}a^kz^{n-k}\tag{1}
\end{align*}

We obtain by successively applying (1)
\begin{align*}
\color{blue}{[x^2yz]}&\color{blue}{(2x-y+z+1)^7}\\
&=[yz]\binom{7}{2}2^2(-y+z+1)^5\\
&=[z]\binom{7}{2}2^2\binom{5}{1}(-1)(z+1)^4\\
&=\binom{7}{2}2^2\binom{5}{1}(-1)\binom{4}{1}\\
&\,\,\color{blue}{=-1\,680}
\end{align*}
in accordance with OPs result.

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