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$2{\mkern 1mu} {\mkern 1mu} \le {\mkern 1mu} {\mkern 1mu} r{\mkern 1mu} {\mkern 1mu} \le {\mkern 1mu} {\mkern 1mu} n{\mkern 1mu} $ for ${\rm{\{ }}{r^n}{\rm{\} }}{\mkern 1mu} + {\mkern 1mu} 2{\mkern 1mu} \left( {\begin{array}{*{20}{c}}
n\\
{r{\mkern 1mu} - {\mkern 1mu} 1}
\end{array}} \right){\mkern 1mu} + {\mkern 1mu} \left( {\begin{array}{*{20}{c}}
n\\
{r{\mkern 1mu} - {\mkern 1mu} 2}
\end{array}} \right){\mkern 1mu} = {\mkern 1mu} .....$
$\left( {\begin{array}{*{20}{c}}{n\, + \,1}\\{r\, - \,1}\end{array}} \right)$
$\,2\,\left( {\begin{array}{*{20}{c}}{n\, + \,1}\\{r\, + \,1}\end{array}} \right)\,$
$\left( {\begin{array}{*{20}{c}}{n\, + \,1}\\r\end{array}} \right)$
$\,\left( {\begin{array}{*{20}{c}}{n\, + \,2}\\r\end{array}} \right)$
Solution
$\left( {\,_r^n} \right)\,\, + \,2\,\left( {\begin{array}{*{20}{c}}
n \\
{r\, – \,1}
\end{array}} \right)\, + \,\left( {\begin{array}{*{20}{c}}
n \\
{r\, – \,2}
\end{array}} \right)$
$ = \,\left[ {\left( {_r^n} \right)\, + \,\left( {\begin{array}{*{20}{c}}
n \\
{r\, – \,1}
\end{array}} \right)\,} \right]\, + \,\left[ {\left( {\begin{array}{*{20}{c}}
n \\
{r\, – \,1}
\end{array}} \right)\,\, + \,\left( {\begin{array}{*{20}{c}}
n \\
{r\, – \,2}
\end{array}} \right)\,} \right]$
$ = \left( {\begin{array}{*{20}{c}}
{n\, + \,1} \\
r
\end{array}} \right)\,\, + \,\left( {\begin{array}{*{20}{c}}
{n\, + \,1} \\
{r\, – \,1}
\end{array}} \right) = \,\left( {\begin{array}{*{20}{c}}
{n\, + \,2} \\
{r\,}
\end{array}} \right)\,$
