$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,{\rm e}^{#1}\,}
\newcommand{\fermi}{\,{\rm f}}
\newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}
\newcommand{\half}{{1 \over 2}}
\newcommand{\ic}{{\rm i}}
\newcommand{\iff}{\Longleftrightarrow}
\newcommand{\imp}{\Longrightarrow}
\newcommand{\pars}[1]{\left(\, #1 \,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\pp}{{\cal P}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,}
\newcommand{\sech}{\,{\rm sech}}
\newcommand{\sgn}{\,{\rm sgn}}
\newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}}
\newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$
$\ds{{n \choose k} = {n - 2 \choose k} + 2{n - 2 \choose k - 1}
+{n - 2 \choose k - 2}:\ {\large ?}}$.
\begin{align}&\color{#66f}{\large%
{n - 2 \choose k} + 2{n - 2 \choose k - 1} + {n - 2 \choose k - 2}}
\\[3mm]&=\oint_{\verts{z}\ =\ 1}\bracks{%
{\pars{1 + z}^{n - 2} \over z^{k + 1}}
+2\,{\pars{1 + z}^{n - 2} \over z^{k}}
+{\pars{1 + z}^{n - 2} \over z^{k - 1}}}\,{\dd z \over 2\pi\ic}
\\[3mm]&=\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{n - 2} \over z^{k + 1}}
\pars{1 + 2z + z^{2}}\,{\dd z \over 2\pi\ic}
\\[3mm]&=\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{n} \over z^{k + 1}}
\,{\dd z \over 2\pi\ic} = \color{#66f}{\large{n \choose k}}
\end{align}