{flalign} not working with function and summations

Question

I'm a newcomer to Latex and have been relying on lots of examples to learn how to format a math proof. I have been really stuck on this particular part of the proof and was unable to find resources or a similar example to fix:

1. the alignment of the equal signs
2. (maybe not an issue) the "g" in the first line is not the same style/font as other "g"'s in the document that are \displaystyle
3. the text label "(multinomial theorem)" should be right-aligned, hopefully this will fix itself when (1) is fixed?

\begin{flalign*}
&& g(\partial x)f &= \sum_{i=1}^{r} a_i(\sum_{j=1}^{n} {c_i}_j\partial_j)^{d}f \\
&= \sum_{i=1}^{r} a_i (\sum_{j=1}^{n} {c_i}_j \partial_j)^{d}f \\
&= \sum_{i=1}^{r} a_i [\sum_{\substack{K=(k_1 ... k_n) \\ |K| = d}}{d \choose k}\prod_{j=1}^{n} ({c_i}_j \partial_j)^{k_j}]f && \text{(multinomial theorem)} \\
\end{flalign*}

Any help is greatly appreciated, thank you!

Answer 1

Simpler is not to use flalign* but simple align*:

\documentclass{article}
\usepackage{amsmath}

%---------------- show page layout. don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%
\begin{document}
    \begin{align*}
g(\partial x)f
    & = \sum_{i=1}^{r} a_i\biggl(\sum_{j=1}^{n} {c_i}_j\partial_j\biggr)^{d}f   \\
    &= \sum_{i=1}^{r} a_i\biggl(\sum_{j=1}^{n} {c_i}_j \partial_j\biggr)^{d}f   \\
    &= \sum_{i=1}^{r} a_i
        \biggl[\sum_{\substack{K=(k_1 ... k_n) \\
                              |K| = d}}
              \binom{d}{k}\prod_{j=1}^{n} ({c_i}_j \partial_j)^{k_j}
        \biggr]f
      \tag{multinomial theorem}
    \end{align*}
\end{document}

gives:

(red lines indicate page layout)

Answer 2

flalign works perfectly fine if, as indicated above in the comment, you drop the initial && whose purpose I do not understand.

\documentclass{article} 
\usepackage{amsmath} 
\begin{document} 
\begin{flalign*} 
 g(\partial x)f &= \sum_{i=1}^{r} a_i(\sum_{j=1}^{n} {c_i}_j\partial_j)^{d}f \\
 &= \sum_{i=1}^{r} a_i (\sum_{j=1}^{n} {c_i}_j \partial_j)^{d}f \\ 
 &= \sum_{i=1}^{r} a_i 
 [\sum_{\substack{K=(k_1 \dots k_n) \\ |K| = d}}{\binom{d}{k}}
 \prod_{j=1}^{n} ({c_i}_j \partial_j)^{k_j}]f && \text{(multinomial theorem)} 
\end{flalign*} 
\end{document}