$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\dd}{{\rm d}}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\ic}{{\rm i}}% \newcommand{\imp}{\Longrightarrow}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert #1 \right\vert}% \newcommand{\yy}{\Longleftrightarrow}$ \begin{align} \pars{1 + x}^{R + M}&=\sum_{k = 0}^{R + M}{R + M \choose k}x^{k} \\ \pars{1 + x}^{R}\pars{1 + x}^{M} &=\sum_{\ell = 0}^{R}{R \choose \ell}x^{\ell}\sum_{\ell' = 0}^{M}{M \choose \ell'}x^{\ell'}\sum_{k = 0}^{R + M}\delta_{k,\ell + \ell'} \\[3mm]&= \sum_{k = 0}^{R + M}x^{k}\sum_{\ell = 0}^{R}{R \choose \ell} \sum_{\ell' = 0}^{M}{M \choose \ell'}\delta_{\ell',k - \ell} = \left.\sum_{k = 0}^{R + M}x^{k}\sum_{\ell = 0}^{R}{R \choose \ell} {M \choose k - \ell}\right\vert_{0 \leq k - \ell \leq M} \end{align}\begin{align} \pars{1 + x}^{R + M}&=\sum_{k = 0}^{R + M}{R + M \choose k}x^{k} \\ \pars{1 + x}^{R}\pars{1 + x}^{M} &=\sum_{\ell = 0}^{R}{R \choose \ell}x^{\ell}\sum_{\ell' = 0}^{M}{M \choose \ell'}x^{\ell'}\sum_{k = 0}^{R + M}\delta_{k,\ell + \ell'} \\[3mm]&= \sum_{k = 0}^{R + M}x^{k}\sum_{\ell = 0}^{R}{R \choose \ell} \sum_{\ell' = 0}^{M}{M \choose \ell'}\delta_{\ell',k - \ell} \\[3mm] & = \left.\sum_{k = 0}^{R + M}x^{k}\sum_{\ell = 0}^{R}{R \choose \ell} {M \choose k - \ell}\right\vert_{0 \leq k - \ell \leq M} \end{align}
$$\color{#0000ff}{\large% {R + M \choose k} = \sum_{\ell = 0 \atop {\vphantom{\LARGE A^{a}}k - M \leq\ \ell\ \leq k}} ^{R}{R \choose \ell}{M \choose k - \ell}}\,, \qquad 0 \leq k \leq R + M $$